Draft...11 aCorresponding author: Tzeng Yih Lam; School of Forestry and Resource Conservation, National Taiwan 12 University, No. 1, Sec. 4, Roosevelt Road, Taipei, 10617, Taiwan;

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Close-range Photogrammetry with Spherical Panoramas for Mapping Spatial Location and Measuring Diameters of Trees

under Forest Canopies

Journal: Canadian Journal of Forest Research

Manuscript ID cjfr-2018-0430.R2

Manuscript Type: Article

Date Submitted by the Author: 11-Mar-2019

Complete List of Authors: Lu, Mei-Kuei; National Taiwan University, School of Forestry and Resource ConservationLam, Tzeng Yih; National Taiwan University, School of Forestry and Resource ConservationPerng, Bo-Hao; National Taiwan University, School of Forestry and Resource ConservationLin, Ho-Tung; National Taiwan University, School of Forestry and Resource Conservation

Keyword: dendrometer, optical fork effects, panoramic photography, terrestrial photogrammetry, forest measurement

Is the invited manuscript for consideration in a Special

Issue? :Not applicable (regular submission)

https://mc06.manuscriptcentral.com/cjfr-pubs

Canadian Journal of Forest Research

Draft

Close-range Photogrammetry with Spherical Panoramas for Mapping Spatial Location… p. 1

1 Close-range Photogrammetry with Spherical Panoramas for Mapping Spatial

2 Location and Measuring Diameters of Trees under Forest Canopies

3 Mei-Kuei Lu1, Tzeng Yih Lam1,a, Bo-Hao Perng1, Ho-Tung Lin1

4

5 1School of Forestry and Resource Conservation, National Taiwan University, No. 1, Sec. 4, Roosevelt Road,

6 Taipei, 10617, Taiwan

7

8 Email: (1) Mei-Kuei Lu ([email protected]); (2) Tzeng Yih Lam ([email protected]); (3) Bo-Hao

9 Perng ([email protected]); (4) Ho-Tung Lin ([email protected])

10

11 aCorresponding author: Tzeng Yih Lam; School of Forestry and Resource Conservation, National Taiwan

12 University, No. 1, Sec. 4, Roosevelt Road, Taipei, 10617, Taiwan; Tel: +886 2 3366 4624; Fax: +886 2 2365

13 4520; Email: [email protected]

14

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15 Abstract

16 Measurement of tree attributes is important to collect information for forest management.

17 Close-range photogrammetry with spherical panoramas has seen very little development

18 and applications compared to aerial photography. This study develops methods to extract

19 azimuth, horizontal distance, diameter at breast height, and upper stem diameters of

20 individual trees from spherical panoramas based on: (1) the trigonometry principle (TRIGO),

21 (2) the TRIGO corrected for terrain slope (TRIGOSLP), and (3) the pinhole camera model

22 (PINHOLE). Twenty three horizontal point sample plots were randomly established in

23 plantations in Taiwan with a sample of 486 trees. Results showed that tree azimuth was

24 accurately and precisely estimated. TRIGO performed the worst in accuracy and precision

25 for all other tree attributes. TRIGOSLP improved the results of TRIGO but had large

26 estimation errors. PINHOLE achieved the best overall precision for all other tree attributes,

27 but was slightly inaccurate for estimating upper stem diameters. PINHOLE requires

28 approaching a tree to attach a target of known size but has the ability to extract an almost

29 continuous set of upper stem diameters from the tree, which could improve estimation of

30 tree volume. Thus, PINHOLE could potentially be an alternative measurement system for

31 hard-to-measure tree attributes.

32 Keywords: dendrometer, optical fork effects, panoramic photography, terrestrial

33 photogrammetry, forest measurement.

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34 1. Introduction

35 As Kershaw et al. (2016; p. 2) aptly put it, “You can’t efficiently make, manage, or study

36 anything you don’t locate and measure.” Thus, forest measurement is an integral part of

37 forest resource management. Tree diameter at breast height (DBH) is the most fundamental

38 individual tree attribute measured in a forest inventory (Lam et al. 2017). It is easily

39 accessible and can be precisely measured with a range of readily available tools (Clark et al.

40 2000a). Furthermore, upper stem diameters are an integral part of many tree volume

41 models and are important for timber production and processing chains (Newnham et al.

42 2015). However, they are rarely obtained because direct measurements by tree climbing

43 and felled tree stem analysis are labor intensive (Larsen 2006) and indirect measurements

44 with the Spiegel Relaskop or the Criterion RD 1000 are expensive and also low in accuracy

45 (Kershaw et al. 2016; pp. 103-104). Many advanced growth and yield models use tree

46 spatial location as an input. However, tree spatial locations are rarely collected as current

47 ground mapping methods are either costly or prone to inaccuracy (Dick et al. 2010). Close-

48 range photogrammetry with spherical panoramas has the potential to be a measurement

49 system to collect data on the hard-to-measure tree attributes.

50 Early works in forestry on close-range photogrammetry include Bartorelli and Cantiani

51 (1962) in developing a “stereodendrometer” with a pair of cameras to measure tree

52 diameters and heights, Grosenbaugh (1963) in presenting theoretical basis of optical

53 dendrometers, and Ashley and Roger (1969) in measuring upper stem diameters of a tree

54 from a 35 mm camera attached to a custom-made frame. More recently, Larsen (2006)

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55 reconstructed a 3-dimensional model of a tree stem from a series of close-up photographs

56 taken at various angles surrounding the tree. Instead of photographing single trees, a

57 panorama that captures extensive representation of an environment could be used to

58 estimate attributes of a cluster of trees. Dick et al. (2010) mapped spatial location of sample

59 trees for major forest types in northwestern New Brunswick, Canada. Fastie (2010)

60 examined precision and accuracy in estimating stand basal area in birch forests of Alaska,

61 USA. Both studies used horizontal panoramas that had 360° horizontal field of view (FOV)

62 but limited vertical FOV. Alternatively, a spherical panorama has 360° horizontal and 180°

63 vertical FOVs that captures the full view of an environment. Thus, it has the benefits of

64 measuring upper stem diameters beyond what could be seen by a horizontal panorama.

65 While panoramic spherical photogrammetry has been applied in many other disciplines

66 such as architecture (Fangi and Nardinocchi 2013), it has not been explored in forestry with

67 the exception of Perng et al. (2018).

68 Past decades have seen significant advancement in the production of inexpensive consumer

69 grade cameras capable of producing high-quality images, cheap digital storage media, and

70 vast computing power (Dick et al. 2010). Spherical panoramas can now be easily produced

71 and stored. Panoramic spherical photogrammetry could be an attractive option for

72 measuring tree attributes in a forest inventory program in terms of cost, accessibility, and

73 convenience. The goal of this study was to explore the potential and limitation of spherical

74 panoramas for estimating tree attributes. In particular, the objectives were to: (1) develop

75 photogrammetry methods for extracting azimuth, horizontal distance, DBH, and upper stem

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76 diameters of individual trees from spherical panoramas, and (2) assess their accuracy and

77 precision.

78 2. Photogrammetry Methods

79 The origin of a spherical panorama with equirectangular projection is located at the upper

80 left corner (Figure S4; Supplementary Materials). It has a dimension of L × H and a FOV of 2π

81 × π, where L is the length in pixels and H is the height in pixels (Figure S4a). Each point in the

82 panorama is defined by a pixel coordinate of (X,Y). The Y-pixel coordinate of the equator of

83 the panorama is defined as Y = H/2 (Figure S4b).

84 Three photogrammetry methods are developed based on: (1) the trigonometry principle

85 (TRIGO), (2) the TRIGO corrected for terrain slope (TRIGOSLP), and (3) the pinhole camera

86 model (PINHOLE). The three methods are used to estimate horizontal distance, DBH, and

87 upper stem diameter at height h (Dh) of a tree in the panorama. Horizontal distance in this

88 study is specifically defined as the horizontal distance from a plot center to the middle and

89 on the face of a tree bole. Azimuth of a tree from a plot center is estimated by the method

90 in Dick et al. (2010) and not by the three methods. Lastly due to optical fork effects, the

91 method by Grosenbaugh (1963) is adapted to correct the estimated tree diameters.

92 2.1 Azimuth (AZ)

93 According to Dick et al. (2010), a panorama with equirectangular projection has a constant

94 horizontal radial scale (° per pixel) of . To estimate azimuth of a tree from the 360RS L

95 plot center, the azimuth of the first sample tree needs to be obtained in the field (Dick et al.

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96 2010). Let T1 be the first sample tree and Tk be the k-th sample tree in the panorama

97 (Figure S4b), the azimuth of the k-th tree is estimated as (AZ, °)

98 (1)Tk T1 1kAZ AZ P RS

99 where, AZT1 is the measured azimuth of the first tree (°), and P1k is the number of pixels

100 between middle of the first tree and middle of the k-th tree in the panorama (Figure S4b).

101 2.2 Trigonometry Principle (TRIGO)

102 TRIGO assumes that the ground between the camera and the sample trees is perfectly level.

103 Although it may not be a realistic assumption, TRIGO is easily implemented in the field. One

104 simply sets up a camera system at the plot center of a spherical panorama shoot without

105 any additional measurement. Defining the height of the camera above ground as M (cm),

106 horizontal distance to a tree is estimated by the trigonometry of a right angled triangle

107 (Figure 1a). The base of the tree must first be identified in the panorama such as point B

108 with pixel coordinate (XB,YB) in Figure S4b. Then, horizontal distance is estimated as (HDIST,

109 m)

110 (2)tan 100BHDIST M

111 where, θB is the angle from the zenith to the tree base (point B) in radians (Figure 1a). A

112 spherical panorama with equirectangular projection has also a constant vertical radial scale

113 (rad per pixel), thus, θB is estimated as . B BY H

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114 Estimation of a tree diameter requires establishing a distance scale (DS, cm per pixel) for all

115 pixels associated to a sample tree. With the assumption of perfectly level ground, the

116 number of pixels between the equator of the panorama (H/2) and the base (point B) of a

117 tree in the panorama corresponds to M on the ground. Thus, DS for TRIGO is

118 . DBH and Dh uncorrected for optical fork effects are estimated 2TRIGO BDS M Y H

119 as (DIAuncor, cm)

120 (3),uncor TRIGO DIA TRIGODIA P DS

121 where, PDIA is the number of pixels of the diameter in the panorama (Figure S4b).

122 2.3 TRIGO Corrected for Terrain Slope (TRIGOSLP)

123 TRIGOSLP relaxes the assumption of perfectly level ground of TRIGO and accounts for trees

124 on sloped terrain. The method requires direct field measurement of terrain slope that a tree

125 is on, which is an additional field effort compared to TRIGO. Let θS be the slope of the

126 terrain that a sample tree is on in radians: θS is negative if the tree is downslope from a plot

127 center and is positive otherwise (Figure 1b). The Law of Sines is first applied to estimate the

128 sloped distance from the plot center to the tree, and the estimated sloped distance is

129 projected to the horizontal distance by the trigonometry of a right angled triangle (Figure

130 1b). In short, HDIST is estimated as

131 (4) sin cos 100sinB S

C

MHDIST

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132 where, θC is the angle between the two lines from the tree base to the camera and to the

133 base of the tripod, respectively, expressed in radians and is estimated as

134 . 2C B S

135 Similar to TRIGO, a distance scale is established to estimate tree diameters taking into

136 account the terrain slope. Thus, DS for TRIGOSLP is , where 2TRIGOSLP B SDS M Y H P

137 PS is the terrain slope in pixels and is estimated as . PS is negative if the tree S SP H

138 is downslope from the plot center and is positive otherwise. Uncorrected DBH and Dh are

139 estimated as

140 (5),uncor TRIGOSLP DIA TRIGOSLPDIA P DS

141 2.4 Pinhole Camera Model (PINHOLE)

142 Dick et al. (2010) proposed a method of mapping tree spatial location from panoramas by

143 attaching targets of known width and length made from printed cardboards on sample

144 trees. These targets are a way of scaling the trees, but additional field effort is needed to

145 approach the sample trees and to attach the targets. Their method is essentially based on

146 the principle of the pinhole camera model (Figure 1c). Thus, HDIST is estimated as

147 (6)100

WHDIST fw

148 where, f is the focal length of the camera lens (mm), W is the width of a target (cm), and w

149 is the width of the target on the camera sensor (mm) (Figure 1c). The w is estimated as

150 , where PW is the width of the target in the panorama in pixels, and IR is the W Rw P I

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151 sensor-length-to-photo-length ratio (mm/pixel), which is the ratio of the sensor format of

152 the camera used (mm) to the dimension of a single photograph taken by the sensor (pixels).

153 A distance scale for PINHOLE is established as . Uncorrected DBH and Dh PINHOLE WDS W P

154 are estimated as

155 (7),uncor PINHOLE DIA PINHOLEDIA P DS

156 2.5 Optical Fork Effects

157 The three photogrammetry methods for estimating tree diameters are a form of optical fork

158 (Grosenbaugh 1963). Optical fork effects occur when two lines of sight (LoS) originating

159 from a camera intersect two opposite edges of a tree stem and form a chord within the tree

160 stem cross-section. Since the chord does not pass through the center of the tree stem, it is

161 not the desired diameter and needs to be corrected to obtain the true diameter. Moreover,

162 when the two LoS intersect the tree stem at a tilt angle such as viewing an upper stem

163 diameter, both form an elliptical plane within the tree stem further complicating the

164 correction (Grosenbaugh 1963). Thus, a general correction method is needed to correct an

165 estimated tree diameter for the optical fork effects from any photogrammetry method and

166 at any tree height.

167 The correction method in this study is adapted from Grosenbaugh (1963), and details of its

168 derivation are provided in the Supplementary Materials and are depicted in Figure S1. Let

169 α” be the fork angle formed by two LoS originating from a camera, and ϕ be the tilt angle of

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170 the camera. Thus, the estimated DBH and Dh corrected for the optical fork effects are (DIA,

171 cm)

172 (8)

1tan 2

cos arctancos

uncorDIA DIA

173 where, the fork angle α” is estimated as , and the tilt angle ϕ is estimated 2DIAP L

174 as with YDIA defined as the Y-pixel coordinate of a point located on 2 DIAH Y H

175 the tree diameter in the panorama (e.g., YU of Point U in Figure S4b). The left term of the

176 right-hand side of Equation (8) is the correction term. Relationships between the correction

177 term, the fork angle α”, and the tilt angle ϕ are presented in Figure S2 (Supplementary

178 Materials). Lastly, only DIA is reported in this study.

179 3. Materials and Methods

180 3.1 Study Location

181 The study was carried out in Heshe Nature Education Area (Heshe; N 23.59241° and E

182 120.88797°) and Xitou Nature Education Area (Xitou; N 23.66700° and E 120.79557°)

183 located at the National Taiwan University Experimental Forest, Taiwan in 2017. A total of 23

184 plots were randomly established with 7 plots in Heshe and 16 plots in Xitou. In Heshe, 3

185 plots were in a Cunninghamia lanceolate (Lamb.) Hook stand with a current mean stand

186 density of 460 trees ha-1 and an average slope of 4.2°, and 4 plots were in a Zelkova serrata

187 (Thunb.) Makino stand planted in 1937 with a current mean stand density of 530 trees ha-1

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188 and an average slope of 4.8°. In Xitou, all 16 plots were established in 16 Cryptomeria

189 japonica (L.f.) D.Don stands with one plot per stand. One C. japonica stand was planted in

190 1914 with a current mean stand density of 310 trees ha-1 and an average slope of 10.1°. The

191 other 15 C. japonica stands were established as a spacing trial experiment in 1950 with

192 current mean stand densities from 280 to 790 trees ha-1 and average slopes from 4.0 to

193 11.7°.

194 3.2 Field Measurement

195 At a plot center, horizontal point sampling (Bitterlich 1984) with an angle gauge with a basal

196 area factor (BAF) of 2 M (i.e., 2 m2ha-1 per tree tallied) was used to select sample trees for

197 measurement. Tally began at north and continued clockwise with the first tallied tree

198 closest to the north azimuth (0°). For a tallied tree, its DBH (to the nearest 0.1 cm) was

199 measured with a caliper with the beam of the caliper facing the plot center. Horizontal

200 distance was measured with Haglöf Vertex IV ultrasonic hypsometer to the nearest 0.01 m.

201 Its azimuth (to the nearest °) was measured with Haglöf HCH electronic compass. A yellow

202 ribbon was tied to the tallied tree at breast height. A printed cardboard measuring 21.0 ×

203 28.0 cm (W = 21.0 cm) was nailed right above the yellow ribbon. A subsample of 8-10 trees

204 were selected with a larger BAF of 6-10 M for measurement of upper stem diameters. For a

205 subsample tree, its upper stem diameters at 2 and 3 m above ground (D2 and D3) were

206 measured with a diameter tape (to the nearest 0.1 cm) and were flagged with yellow

207 ribbons. Labeling trees with the yellow ribbons was only necessary in this study so that the

208 measured diameters on the sample trees could be accurately identified during image

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209 postprocessing for assessing accuracy and precision of a photogrammetry method. Lastly,

210 terrain slope (to the nearest °), defined as slope from the plot center to the center of the

211 tallied tree at ground level, was only measured in 9 plots.

212 A spherical panorama was shot at the plot center with the full frame Canon EOS 5D Mark III

213 camera with Canon EF 20 mm f/2.8 lens (f = 20 mm) mounted on an automated panoramic

214 head Roundshot VR Drive II (Seitz Phototechnik AG, Lustdorf, Switzerland) in portrait

215 orientation. The camera has a sensor format of 36 × 24 mm. The photographs taken with

216 22.1 megapixel resolution at 3:2 aspect ratio have a dimension of 5760 × 3840 pixels. Thus,

217 the sensor-length-to-photo-length ratio (IR) for our camera system was 0.00625 mm/pixel.

218 The entrance pupil of the camera lens was set at 1.3 m above ground (M = 130 cm). The

219 camera system was set up such that the camera’s nodal point was directly above the

220 rotational axis of the tripod to reduce parallax and distortion during panorama stitching. A

221 total of 60 photographs with at least 45% overlap between two adjacent photographs were

222 shot at 45° vertical and at 30° horizontal intervals.

223 3.3 Image Postprocessing

224 Each set of 60 photographs was stitched into a spherical panorama with equirectangular

225 projection using the software Kolor Autopano Giga (Kolor SARL, Francin, France). The

226 spherical panorama image was imported into QGIS (http://qgis.org) for feature extraction. A

227 point feature shapefile was created to store plot numbers, tree numbers, X- and Y-pixel

228 coordinates of points, camera heights, and heights of measured diameters. Details of

229 feature extraction for a tallied tree in the panorama are described in full in the

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230 Supplementary Materials and depicted in Figure S3. For a tallied tree, either 5 or 9 point

231 features were extracted depending whether the tree was selected for upper stem diameter

232 measurement: (1) 2 point features on the opposite edges of the yellow ribbon at DBH, (2) 1

233 point feature on the middle of the tree base, (3) 2 point features on the opposite edges of

234 the top of the printed cardboard, and (4) 4 point features on the opposite edges of the

235 yellow ribbons at D2 and at D3 (Figure S3). An example of pixel coordinates of point

236 features extracted for a tallied tree and calculations of the parameters of the three

237 photogrammetry methods is presented in the Supplementary Materials and Figure S3.

238 3.4 Data Analysis

239 A total of 618 trees were tallied from the 23 plots. Only 486 trees (78.6%) were fully visible

240 in the panoramas and were used in the analyses. Out of the 486 trees, 182 trees were

241 selected for upper stem diameter measurement, 225 trees were measured for terrain slope,

242 and 78 trees were measured for both terrain slope and upper stem diameters. So, for TRIGO

243 and PINHOLE, 486 trees were analyzed for azimuth, DBH, and horizontal distance, and 182

244 trees were analyzed for upper stem diameters. For TRIGOSLP, it was 225 and 78 trees,

245 respectively.

246 Let be an estimated tree attribute of the k-th tallied tree from a photogrammetry 𝛿𝑘

247 method, and be the measured tree attribute of the k-th tree. Estimation error (E) was k

248 defined as , whereby a negative value suggested underestimation and a 𝐸𝑘 = 𝛿𝑘 ― 𝛿𝑘

249 positive value otherwise. Percent error (PE, %) was calculated as . 𝑃𝐸𝑖 = 100 × (𝛿𝑖 ― 𝛿𝑖) 𝛿𝑖

250 Mean error (ME), mean percent error (MPE), root-mean-squared error (RMSE), and root-

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251 mean-squared percent error (RMSPE) were , , 1

nii

ME E n

1

nii

MPE PE n

252 , and , respectively, where n 2

1

nii

RMSE E ME n

2

1

nii

RMSPE PE MPE n

253 was the number of tallied trees for an analysis.

254 Following Dick et al. (2010), the spatial error of the k-th tallied tree was assessed. The X-

255 and Y-coordinate of a tallied tree could be computed from its azimuth and horizontal

256 distance using the polar coordinate system as and , cosX HDIST sinY HDIST

257 where, γ was the angular coordinate in the polar coordinate system; here γ was converted

258 from the azimuth and had a value of [0, π] in a counterclockwise direction from east to west

259 and a value of [0, -π) in a clockwise direction from east to west. For a tallied tree, its

260 observed X-Y coordinates in the field were calculated from its field measured horizontal

261 distance and azimuth (Xmeas, Ymeas). Its predicted X-Y coordinates were calculated using the

262 horizontal distance and azimuth estimated from the photogrammetry methods (Xpred, Ypred).

263 Thus, the spatial error for the k-th tallied tree was defined as the distance between the two

264 X-Y coordinates as . Mean and root-mean- 2 2

, , , , ,spatial k pred k meas k pred k meas kE X X Y Y

265 squared of Espatial were computed for each photogrammetry method.

266 4. Results

267 4.1 Spatial location

268 Tree azimuth was on average accurately and precisely estimated (ME ± RMSE = 2.69° ±

269 3.30°; Table 1). MPE and RMSPE were not calculated because azimuth is a direction and

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270 does not have a magnitude. The error distributions appeared to correlate with the

271 measured azimuth in a sinusoid trend on average and for each plot (Figure 2). Large positive

272 errors occurred around 90° and 270° azimuths, while large negative errors occurred around

273 0° and 180° azimuths. Since azimuth is a direction, a circular regression (Sarma and

274 Jammalamadaka 1993) was fitted to predict the measured azimuth from the estimated

275 azimuth with the circular package (Agostinelli and Lund 2017) in R (R Core Team 2018). The

276 estimated and measured azimuths were highly correlated with the proportion of explained

277 variance ρ = 0.9986, and the residual standard error of the fitted model was 3.505° (Figure

278 S5; Supplementary Materials).

279 For HDIST, TRIGO had the highest ME and MPE and the largest RMSE and RMSPE (30.24 ±

280 234.96 m, 278.41 ± 1959.73%), while PINHOLE had the lowest ME and MPE and the smallest

281 RMSE and RMSPE (-0.90 ± 0.72 m, -9.60 ± 3.31%) (Table 1). TRIGO’s performance was

282 heavily influenced by extreme errors ranging from 500 m to 4600 m (Figure 3a). TRIGOSLP

283 significantly improved the accuracy and precision of TRIGO by reducing the extreme errors

284 to about 33 m (Figure 3b). For TRIGO and TRIGOSLP, the funnel shape in the distributions of

285 the errors and the increasing trend in the LOESS regressions suggested that variability and

286 average error increased with larger observed HDIST (Figure 3a-b). On the contrary, PINHOLE

287 produced a consistently decreasing trend in the distribution of the errors without a distinct

288 funnel shape suggesting that PINHOLE could precisely estimate HDIST but increasingly

289 underestimated it for trees far from a plot center (Figure 3c). However, the LOESS

290 regression of the errors in percentage against observed HDIST between 0 and 18 m

291 suggested that PINHOLE produced similar relative errors for trees within these distances

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292 (Figure S6c; Supplementary Materials). Lastly for several trees, PINHOLE produced large

293 errors up to -9 m (Figure 3c).

294 It was without surprise that spatial error was the largest for TRIGO and the least for

295 PINHOLE, with TRIGOSLP in between (Table 1). However, spatial distributions of Espatial

296 differed between the three methods (Figure S7; Supplementary Materials). For TRIGO and

297 TRIGOSLP, sample trees located at the southeast and southwest azimuths generally had

298 some of the largest Espatial (Figure S7a-b), but Espatial of PINHOLE was more homogenously

299 distributed across azimuths and was clearly positively correlated with observed distance

300 from a plot center (Figure S7c). As Espatial represented a combined horizontal accuracy of the

301 errors in estimating azimuth and horizontal distance (Dick et al. 2010), these results pointed

302 out that error in AZ had a smaller effect than error in HDIST on Espatial.

303 4.2 Tree diameters

304 Similar to HDIST, TRIGO consistently produced the highest ME and MPE and the largest

305 RMSE and RMSPE for DBH, D2, and D3 (Table 1). For DBH, PINHOLE was the most accurate

306 with ME and MPE about eight to ten times less than TRIGOSLP (Table 1). For D2 and D3,

307 TRIGOSLP was the most accurate with ME and MPE about two times less than PINHOLE

308 (Table 1). Nonetheless, PINHOLE was consistently the most precise method for all three

309 diameters.

310 Results of the distributions of the errors from TRIGO for DBH, D2, and D3 over their

311 respective observed values resembled those of HDIST with distinct funnel shapes and with

312 extreme errors (Figure 4). TRIGOSLP on average slightly overestimated DBH of larger trees

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313 (Figure 4b) but was on average accurate for D2 and D3 despite large variability across the

314 observed values (Figure 4e,h). PINHOLE had the least variability in the distributions of the

315 errors over the observed values for DBH, D2, and D3, whereby the error was generally

316 within ± 6 cm (Figure 4c,f,i). However, PINHOLE on average increasingly underestimated

317 diameters of larger trees. Lastly, the distributions of the errors in percentage for DBH, D2,

318 and D3 mirrored these results with the exception that the relative errors of PINHOLE were

319 similar between small and large trees (Figure S8; Supplementary Materials).

320 Distributions of the errors for DBH, D2, and D3 over observed HDIST generally resembled

321 the above results with the exception of PINHOLE (Figure 5). For example, TRIGOSLP on

322 average overestimated DBH of trees at a greater distance from the plot center (Figure 5b)

323 but consistently underestimated D2 and D3 slightly across observed HDIST (Figure 5e,h). For

324 PINHOLE, the error was generally within ± 6 cm across observed HDIST for the three

325 diameters, but the LOESS regressions showed an improvement in the estimation for trees

326 further from the plot center (Figure 5c,f,i). For example, for DBH, PINHOLE produced large

327 negative average errors for trees less than 4 m from the plot center, but average errors

328 were closed to zero for trees between 9 and 20 m from the plot center (Figure 5c). Lastly,

329 distributions of the errors in percentage of tree diameters over observed HDIST largely

330 mirrored these results for the three methods (Figure S9; Supplementary Materials).

331 5. Discussion

332 This study has explored the use of spherical panoramas in estimating attributes of sample

333 trees under forest canopies with methods based on the trigonometry principles and the

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334 pinhole camera model. Tree azimuth was independently estimated with accuracy and

335 precision comparable to Dick et al. (2010), who have reported an average absolute error of

336 2.3 ± 2.5°. Their study is the only known work that estimates tree azimuth from the

337 panoramas. Dick et al. (2010) used a compass mounted on a tripod to measure azimuth

338 whereas an electronic compass was used in this study. The Haglöf HCH electronic compass

339 has a reported accuracy of 2.5° (Haglöf Sweden AB, Sweden), which may have contributed

340 to the slightly higher estimation error in this study. For all other tree attributes, TRIGO

341 consistently performed the poorest, but accuracy and precision were significantly improved

342 after accounting for terrain slope by TRIGOSLP. PINHOLE consistently achieved the best

343 precision for all tree attributes and the best accuracy for HDIST and DBH, but it was slightly

344 inaccurate for D2 and D3.

345 It is expected that TRIGO performs poorly when ground surface is far from level. The errors

346 stem from |tan(θB)| (Equation 2) and |YB – H/2|, which are the component of the

347 trigonometry of a right angled triangle for estimating HDIST and the component of the

348 distance scale for estimating tree diameters, respectively. For example, for a tree near the

349 plot center and on steep upslope, its tree base will be imaged near the equator of a

350 panorama. As a result, its estimated θB will be close to 90° and |YB – H/2| will be small,

351 which in turn produces larger than expected |tan(θB)| and DSTRIGO that leads to

352 overestimating HDIST and tree diameters. This may explain why TRIGO produced the

353 extreme spatial errors for the sample trees at south to southwest azimuths (Figure S7a).

354 These sample trees were mainly on the upslope from a plot center with some trees on slope

355 greater than 15° (Figure S10; Supplementary Materials). Thus, their |tan(θB)| were likely

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356 overestimated. However, to verify this assumption, a study with trees on flat terrain is

357 necessary. Nonetheless, this study has shown that TRIGO has limited use in forests with

358 complex terrain, but it should be feasible for forests or urban parks with flat terrain. Future

359 studies could explore the potential applications of TRIGO on flat terrain due to its simplicity.

360 TRIGOSLP has significantly improved the accuracy and precision of estimating the tree

361 attributes by accounting for terrain slope, but it still produced large positive errors. For

362 TRIGO and TRIGOSLP, imprecision tends to increase with increasing distance from the plot

363 center, especially for HDIST and DBH, as shown by the funnel shaped distributions of errors.

364 Takahashi et al. (1997) and Perng et al. (2018) have reported similar findings. The funnel

365 shape patterns could be explained by the object space resolution of our camera system.

366 Following Clark et al. (2000b), our camera system has an object space resolution of ± 0.3125

367 mm/m, e.g., diameter can only be recorded in 0.31 cm increment per pixel for a tree at 10

368 m away. Thus, it explains the relationship between precision and distance from plot center.

369 For TRIGO and TRIGOSLP, a major source of errors is the accurate identification of tree base.

370 Ground vegetation and woody debris are the major obstructions. In our experience,

371 occlusion of tree base was less of an issue in the 7 Heshe plots because of little ground

372 vegetation cover (see Figure S3). However, ground vegetation could be up to 50 cm high in

373 all 16 Xitou plots. In this study, we have attempted to clear as much ground vegetation as

374 possible surrounding the sample trees. Fastie (2010) found similar challenges in their work.

375 Dick et al. (2010) found that tree visibility increased as stand density increased likely due to

376 reduction in understory vegetation cover from increasing stand density in their study area.

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377 However, occlusion by ground vegetation is not unique to close-range photogrammetry

378 with spherical panoramas. Maas et al. (2008) found that tree height was underestimated

379 due to occlusion caused by understory vegetation. By integrating airborne and terrestrial

380 laser scanning, Giannetti et al. (2018) found improvement in estimated tree height in

381 Mediterranean forests with dense vegetation. For TRIGO and TRIGOSLP, clearing ground

382 vegetation would improve accuracy in identifying tree base during analyses, but it would

383 substantially add time and cost in stands with dense ground vegetation. Moreover,

384 TRIGOSLP requires the additional effort of measuring terrain slope. Thus, these trade-offs

385 may not be justifiable for the accuracy and precision achieved by TRIGO and TRIGOSLP to be

386 applied in many forest conditions with perhaps the exception of forests with relatively flat

387 terrain cleared of ground vegetation.

388 PINHOLE scales each individual tree with its attached printed cardboard of known size; thus,

389 avoiding the need to identify the tree base for scaling. This establishes a more accurate

390 scale for each tree, and leads to PINHOLE achieving overall higher precision and comparable

391 accuracy to TRIGOSLP. However, PINHOLE produced several large errors for HDIST and D3.

392 These errors were due to issues in stitching out-of-focus photographs. The original

393 photographs showed that the printed targets attached these trees were out of focus, which

394 led to stitching that distorted the trees in the panoramas, and in turn, resulted in scaling

395 errors. This could be easily avoided in the future by checking for out-of-focus photographs

396 in the field. In light of object space resolution, we would expect that: (1) trees of different

397 sizes at the same distance from a plot center would have similar errors in their estimated

398 diameters, and (2) errors in estimated diameters increase with increasing distance from a

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399 plot center. However, PINHOLE produced results contrary to our expectations, especially for

400 DBH. With horizontal point sampling, there is a positive correlation between the size of

401 sample trees and their distances from the plot center (Figure S11; Supplementary

402 Materials). This may explain why estimated diameters of larger trees in this study were less

403 accurate and more variable (Figure 4) as most of them were further away from the plot

404 center, which agrees with the basis of the object space resolution. However, this

405 explanation contradicts the results that diameter estimates of trees closer to the plot center

406 were more negatively biased than trees further from the plot center as depicted in Figure 5.

407 To fully explain the contradictions, we recommend two experiments on flat terrain. One

408 control experiment uses square and cylindrical objects of known sizes and placed at specific

409 distances from a plot center. Another experiment is in a mature and low density forest

410 stand without obstruction from ground vegetation.

411 Several past studies have applied some forms of scaling to estimate tree attributes. Using

412 scales and a 200 mm lens, Crosby et al. (1983) achieved an average error of 0.061 ± 1.91% in

413 estimating upper stem diameters at 10 m distance. Using tree height for scaling, Clark et al.

414 (2000b) produced an error of -0.23 ± 2.4 cm for tree DBH and upper stem diameters with a

415 consumer grade camera Kodak DC-120. Using printed targets and horizontal panoramas

416 similar to our study, Dick et al. (2010) estimated average absolute error of 0.23 ± 0.32 m for

417 horizontal distance. Perng et al. (2018) developed a stereoscopy method with two spherical

418 panoramas vertically displaced at a known distance. Using this distance as scale, Perng et al.

419 (2018) produced average errors of -0.11 ± 3.53 m, -0.77 ± 13.19 cm, -0.63 ± 7.55 cm, and -

420 0.87 ± 7.51 cm for HDIST, DBH, D2, and D3, respectively. In short, the results from the four

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421 studies were more accurate than the corresponding ones in our study. However, the

422 reported precisions in our study were either comparable or better than the four studies.

423 Furthermore, Dick et al. (2010) found that horizontal distance was underestimated for trees

424 at greater distances from a plot center, which agrees with the PINHOLE results in our study.

425 Two potential sources of errors in estimating tree attributes from spherical panoramas

426 common to the three photogrammetry methods are stitching errors and map projection

427 errors. Like most stitching software, Autopano Giga uses propriety algorithms for stitching,

428 and their accuracy against ground truth have not been reported. In our study, at least

429 100,000 control points per spherical panorama were automatically identified by the

430 software, and we manually edited the poor ones prior to stitching. Thus, stitching errors

431 might be minimized using at least 100,000 control points. However, a potential indication of

432 stitching issues is the consistent sinusoidal trends in the estimated tree azimuth. In the field,

433 the camera system always pointed towards north as the starting point, but how this is

434 connected to the sinusoidal trends is not immediately clear. Another potential stitching

435 issue is the curved horizon which could introduce errors in the estimation (Perng et al.

436 2018). In our experience, Autopano Giga generally produces spherical panoramas with

437 leveled horizon. For a very few exceptions which this was not so, we manually added

438 control points where level terrain in the images was to correct for the curved horizon.

439 Nonetheless, to verify potential stitching errors caused by Autopano Giga, we recommend a

440 future study that compares accuracy and precision in the estimated tree attributes between

441 different stitching software; particularly Hugin (hugin.sourceforge.net), which is open

442 source with its source codes available for download and review.

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443 Equirectangular projection is widely used in panoramic spherical photogrammetry because

444 of the simple relationship between the position of a pixel in the image and its

445 corresponding point on the sphere image (Fangi and Nardinocchi 2013). This simple

446 relationship allows us to easily compute the angles needed for the TRIGO, TRIGOSLP, and

447 the optical fork effects correction method. However, equirectangular projections are

448 neither conformal nor equal-area with distortion increasing from the equator to the

449 maximum at the zenith and at the nadir (Snyder 1987; pp. 90-91). This distortion likely has

450 minimum effects on estimated DBH because breast height is usually near the equator of a

451 spherical panorama. Conversely, upper stem diameters are more likely affected, which may

452 have contributed to the lower accuracy in this study. Tree upper bole near the zenith

453 appears to be larger than what is expected, and tree crown appears to be compressed and

454 curved in the panoramas (see Figure S3). Thus, with equirectangular projection, upper stem

455 diameters could only be estimated up to a certain height. Beyond that, tree taper is not

456 preserved. The distortion is also more severe for trees nearer to a plot center. Other map

457 projections such as Mercator projection, Miller Cylindrical projection, and Lambert

458 Cylindrical Equal-Area projection are possible alternatives. The first two are conformal with

459 less distortion towards the zenith, while the third is equal area (Snyder 1987; pp. 38-47, 86-

460 89, 76-85). A future study could first compare different map projections in estimating

461 known width of a man-made structure such as rows of windows on a building and then

462 compare them to estimating upper stem diameters. Another potential study could compare

463 estimated upper stem diameters of the same trees between a spherical panorama and a

464 horizontal panorama. Dick et al. (2010) noted that a horizontal panorama has minimal

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465 distortion with proper field setup and geometric adjustment by a stitching software despite

466 having limited vertical FOV. This comparison would allow one to explore how distortion bias

467 progresses up a tree bole between a spherical and a horizontal panorama.

468 We recommend PINHOLE among the three photogrammetry methods. For one, it is the

469 least affected by ground vegetation. Secondly, it has the best overall precision in estimating

470 tree diameters. We are in favor of trading overall higher precision in PINHOLE for a slight

471 loss in accuracy because this implies that our estimates of tree attributes are more

472 consistent even for trees further from a plot center. Thirdly, PINHOLE could potentially

473 advance methods of measuring upper stem diameters. Past studies have stressed the

474 important role of upper stem diameters in estimating tree volume (Schmid-Haas and

475 Winzeler 1981). Berger et al. (2014) showed that accurate upper stem diameter

476 measurement could lead to improved precision in tree volume estimation. Over the years, a

477 range of instruments have been developed for measuring upper stem diameters. Parker and

478 Matney (1999) compared between Criterion RD 400, Tele-Relaskop, and Wheeler

479 pentaprism in measuring upper stem diameters and reported mean percent accuracy and

480 standard error of -1.92 ± 0.60%, 1.54 ± 0.66%, and -4.16 ± 0.56%, respectively. In another

481 study, Williams et al. (1999) compared Barr & Stroud dendrometer to Criterion RD 400 and

482 reported mean error and standard error of 0.34 ± 0.88 cm and 0.12 ± 1.43 cm, respectively.

483 Despite that their accuracy and precision were better than PINHOLE, they could only

484 measure a few upper stem diameters in the field because of the required efforts. On the

485 other hand, PINHOLE could potentially extract an almost continuous set of upper stem

486 diameters from a tree up to a certain height, and the data could be used to construct tree

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487 taper models or to determine individual tree volume by graphical or integration methods

488 (Kershaw et al. 2016; pp. 144-147). PINHOLE could also be integrated into common

489 sampling designs such as Big BAF sampling (Marshall et al. 2004). For example, under Big

490 BAF sampling, a small BAF is used to count trees and a large BAF is used to select trees for

491 imaging and assessment of tree volume by the PINHOLE method. Given the terrain in our

492 study sites, the process of identifying a sample tree, measuring its DBH, and nailing a

493 printed cardboard at breast height level took approximately five minutes. We believe that

494 these advantages with PINHOLE may outweigh the added effort in the field. Lastly, the total

495 cost of our camera system in this study was approximately USD 10,000 due to the expensive

496 automated panoramic head. The cost will be significantly lower with a cheaper full frame

497 camera and a manual panoramic head, which should be attractive for managers with

498 limited forest inventory budget. Nonetheless, for PINHOLE to be widely applicable, its

499 accuracy and precision needs to be improved possibly through the above recommended

500 studies.

501 6. Conclusion

502 Although remote sensing has a long history in forestry (Lewis 1919; Seely and Seeley 1934),

503 close-range photogrammetry with spherical panoramas for estimating tree attributes under

504 forest canopies have not been extensively explored. Future studies are needed to further

505 minimize errors from PINHOLE such as potentially using lens of longer focal length.

506 Consumer grade cameras that can take a spherical panorama with a press of a button such

507 as Ricoh Theta V (Ricoh Company Ltd, Japan) and Garmin VIRB 360 (Garmin Ltd, USA) are

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508 accessible. Their potentials should be explored in future work, especially understanding

509 their image projection methods that could be an issue to estimation. While this study has

510 demonstrated that close-range photogrammetry with spherical panoramas could

511 potentially be a low-cost alternative to a terrestrial laser scanning system, a future study

512 comparing accuracy and precision in estimating tree attributes between the two systems is

513 needed.

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514 Acknowledgements

515 We especially thank the field crew of their diligence in collecting data under difficult terrain

516 and conditions. We are also especially grateful to the Associate Editor and Jeffrey H. Gove

517 for spending significant amount of time and effort to provide extremely helpful comments

518 and edits that have significantly improved this paper. We also like to especially thank

519 Christie Quon for language editing. This study was supported by the Ministry of Science and

520 Technology (MOST) Taiwan R.O.C. (grant number MOST 105-2628-B-002-010-MY3).

521 Conflict of Interest Statement

522 None declared.

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Tables

Table 1. Mean error (ME), mean percent error (MPE), root-mean-squared error (RMSE), and

root-mean-squared percent error (RMSPE) for tree azimuth (AZ), horizontal distance

(HDIST), spatial error (SPATIAL), diameter at breast height (DBH), and upper stem diameters

at 2 and at 3 m (D2 and D3).

Tree Attribute ME RMSE MPEa RMSPEa

AZ (°) 2.69 3.30 NA NA

HDIST (m)TRIGO 30.24 234.96 278.41 1959.73

TRIGOSLP 1.52 4.17 12.86 32.58PINHOLE -0.90 0.72 -9.60 3.31

SPATIAL (m)TRIGO 33.85 234.47 NA NA

TRIGOSLP 2.43 3.79 NA NAPINHOLE 1.08 0.77 NA NA

DBH (cm)TRIGO 120.93 845.35 275.66 1943.65

TRIGOSLP 5.82 15.07 14.16 32.95PINHOLE -0.66 1.75 -1.34 4.46

D2 (cm)TRIGO 86.81 420.72 189.74 875.11

TRIGOSLP -0.77 5.90 -1.39 13.96PINHOLE -1.57 2.37 -3.78 4.89

D3 (cm)TRIGO 80.38 393.04 188.29 869.46

TRIGOSLP -0.83 5.37 -1.58 13.43PINHOLE -1.63 2.30 -4.08 5.33

aNA = Not calculated.

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Figures

Fig. 1. Photogrammetry methods based on (a) the trigonometry principle (TRIGO), (b) the

TRIGO corrected for terrain slope (TRIGOSLP), and (c) the pinhole camera model (PINHOLE)

for estimating tree diameter (DIA, cm) and horizontal distance from a plot center to the

middle front face of a tree (HDIST, m). The parameters are: M is camera height (cm), θB is

the angle from the zenith to the tree base (rad), θC is the angle between the two lines from

the tree base to the camera and to the base of the tripod, respectively (rad), θS is the terrain

slope angle (rad), f is the focal length of the camera lens (mm), W is the actual width of the

target (cm), and w is the width of the target on the camera sensor (mm).

Fig. 2. Error of estimated azimuth against measured azimuth. Each point depicts a tallied

tree. Thick black solid line depicts fitted locally weighted polynomial (LOESS) regression of

error against measured azimuth. Gray solid lines depict fitted LOESS regression for each

sample plot.

Fig. 3. Error against observed horizontal distance for (a) the trigonometry principle (TRIGO),

(b) the TRIGO corrected for terrain slope (TRIGOSLP), and (c) the pinhole camera model

(PINHOLE). Each point depicts a tallied tree. Black points depict tallied trees with terrain

slope measurement. Gray points depict tallied trees without terrain slope measurement.

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Thick red solid lines depict fitted locally weighted polynomial (LOESS) regression of errors

against observations.

Fig. 4. Error against observed DBH and upper stem diameters at 2 and 3 m (D2 and D3) for

(a,d,g) the trigonometry principle (TRIGO), (b,e,h) the TRIGO corrected for terrain slope

(TRIGOSLP), and (c,f,i) the pinhole camera model (PINHOLE). Each point depicts a tallied

tree. Black points depict tallied trees with terrain slope measurement. Gray points depict

tallied trees without terrain slope measurement. Thick red solid lines depict fitted locally

weighted polynomial (LOESS) regression of error against observations.

Fig. 5. Error of estimated DBH and upper stem diameters at 2 and 3 m (D2 and D3) against

observed horizontal distance for (a,d,g) the trigonometry principle (TRIGO), (b,e,h) the

TRIGO corrected for tree slope (TRIGOSLP), and (c,f,i) the pinhole camera model (PINHOLE).

Each point depicts a tallied tree. Black points depict tallied trees with terrain slope

measurement. Gray points depict tallied trees without terrain slope measurement. Thick

red solid lines depict fitted locally weighted polynomial (LOESS) regression of error against

observed horizontal distance.

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Fig. 1

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Fig. 2

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Fig. 3

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Fig. 4

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Fig. 5

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Documents

Draft...11 aCorresponding author: Tzeng Yih Lam; School of Forestry and Resource Conservation, National Taiwan 12 University, No. 1, Sec. 4, Roosevelt Road, Taipei, 10617, Taiwan;