# Comparative Evaluation of Algorithms for Leaf Area Index Estimation from Digital Hemispherical Photography through Virtual Forests

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## Abstract

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^{2}> 0.8, RMSD < 0.2). After correcting for the vegetation clumping effect, there was a large inconsistency. In general, Can_Eye more accurately estimated true LAI than CIMES and Hemisfer (with R

^{2}= 0.88 > 0.72, 0.49; RMSE = 0.45 < 0.7, 0.94; nRMSE = 15.7% < 24.21%, 32.81%). There was a systematic underestimation of PAI and LAI using Hemisfer. The most accurate algorithm for estimating LAI was identified as the P57 algorithm in Can_Eye which used the 57.5° gap fraction inversion combined with the finite-length averaging clumping correction. These results demonstrated the inconsistency of LAI estimates from DHP using different algorithms. It highlights the importance and provides a reference for standardizing the algorithm protocol for in situ forest LAI measurement using DHP.

## 1. Introduction

_{eff}) is derived from DHP, assuming that canopy elements are randomly distributed, while true PAI (PAI

_{true}) is derived if the non-randomness of canopy elements is corrected through the estimation of the clumping index.

_{eff,}as it does not correct for the clumping effect. Can_Eye, developed by the French National Institute of Agricultural Research, has been used extensively in previous studies [23,24,25,26,27]. Hemisfer, developed by the Swiss Federal Institute for the Forest, Snow, and Landscape Research, has been widely used as well [28,29,30]. CIMES is a package of programs encompassing various LAI retrieval methods and is particularly flexible for batch processing multiple DHP images [31,32,33]. Faced with these options, the question that often arises for users is which algorithm of which program produces more accurate LAI estimates. Addressing this problem can provide guidance for standardizing LAI estimation protocols and reducing in situ LAI measurement uncertainty.

_{eff}estimates [35]. The study by Promis et al. (2011) found similar PAI

_{eff}estimates from GLA and Winphot [36]. Similarly, Hall et al. (2017) found that Can_Eye and CIMES produced comparable PAI

_{eff}estimates. However, the two programs produced statistically significant different clumping index estimates and, thus, different PAI

_{true}estimates [37]. It is worth noting that these studies only evaluated the consistency of results from different algorithms, instead of the accuracy of each algorithm, due to a lack of true reference values. A few studies have used destructive sampling or litterfall collection to acquire LAI reference values for validating the accuracy of DHP in LAI estimation [27,38,39]. However, there is no conclusive evidence concerning the accuracy comparison of various algorithms implemented in these different DHP programs. This calls for an accuracy assessment so that they can be used in the community with confidence.

_{true}estimates [43,44], leaf angle distribution [45], and slope correction on estimating PAI

_{eff}[46]. Nevertheless, these studies used simple geometric primitives to model trees, which differ from real trees, especially in terms of the woody component structure. Recently, highly realistic tree models have been reconstructed from terrestrial LiDAR point clouds and further used to construct virtual forests [47]. Combined with a ray-tracing technique, synthetic DHP is generated for evaluating the retrieval of the clumping index [48]. More recently, Zou et al. (2018) used virtual forests to assess the performance of seven inversion models in estimating the PAI and LAI values from combined leaf-on and leaf-off DHP [49]. Some simulation studies use virtual isolated trees with realistic tree architecture to evaluate the accuracy of leaf area density and LAI estimation for individual trees [50,51]. To the best of our knowledge, there have been no conclusive studies with accuracy evaluations of algorithms implemented in commonly used DHP software programs. Virtual forests provide the potential to solve this problem.

## 2. Materials and Methods

#### 2.1. Virtual Forests Generation

^{2}. Leaves were inserted to the woody QSM model using a revised non-intersecting leaf insertion algorithm (QSM-FaNNI) [54], so that leaves intersected neither other leaves nor other woody components. Trees of different leaf densities were created. In addition, we revised the original QSM-FaNNI leaf insertion method so that the orientation of all leaves followed pre-defined leaf inclination angle distribution types. As a result, we received a collection of 180 highly realistic tree models, examples of which are shown in Figure 2.

_{true-ref}and PA

_{Itrue-ref}hereafter. They were derived directly from the virtual forest stands, by taking the ratio between “half of the total surface areas of all leaves and all woody components in the forest stand” and “the ground area of the forest stand”. To be consistent with the usual height of 1.5 m above ground for most DHP collections, only the leaves and woody components above this height were incorporated in the computation. The stand LAI

_{true-ref}values ranged from 0.52 to 5.53, while the stand PAI

_{true-ref}values ranged from 1.43 to 6.38. In each stand, a circular plot with a radius of 25 m was selected for taking DHP. The plot LAI

_{true-ref}values ranged from 0.49 to 5.39, while the plot PAI

_{true-ref}values ranged from 1.3 to 6.15. The LAI

_{true-ref}and PA

_{Itrue-re}

_{f}of each virtual forest stand are shown in Table 1.

#### 2.2. Synthetic DHP Generation

#### 2.3. LAI Estimation from DHP

_{eff}) and true PAI (PAI

_{true}) can be related to gap fraction using the equations:

_{true}[55], three main methods were proposed to estimate the clumping index ($\lambda \left(\theta \right)$), including the finite-length averaging method (LX) [63], the gap size distribution method (CC) [64,65], and the combination of LX and CC (CLX) method [66]. The finite-length averaging (LX) method was proposed by Lang and Xiang in 1986 using the following equation:

_{true}to LAI

_{true}. In this study, we used leaf-on and leaf-off DHP to estimate PAI and woody area index (WAI), respectively [68]. Then, the LAI was estimated using:

_{true}is the estimate of PAI after considering canopy non-randomness in leaf-on conditions, WAI

_{true}is the estimate of WAI (PAI in leaf-off conditions) after considering canopy non-randomness, and LAI

_{true}is the final estimate of LAI.

_{true}and LAI

_{true}estimates (4 from Can_Eye, 9 from CIMES, and 24 from Hemisfer). The differences among these algorithms were mainly in how the PAI

_{eff}was estimated, how the orientation of leaves and the clumping index were estimated, and how pure canopy segments with zero gap fraction were handled. Summarized descriptions of the 37 algorithms are presented in Table 2, Table 3 and Table 4 for more details.

#### 2.4. Statistical Analysis

_{eff}and LAI

_{eff}, we calculated the consistency among the three programs (Can_Eye, CIMES, and Hemisfer), using the coefficient of determination (R

^{2}), and the root mean square difference (RMSD). Higher values of R

^{2}and lower values of RMSD indicated greater consistency and robustness.

_{true}and LAI

_{true}, the values calculated from the virtual forests as described in Section 2.1 were used as the true reference values. Because the three programs offered 37 estimates of PAI and LAI from different algorithms, we first identified the most accurate results within each software program. In addition, the most accurate results were subsequently used for inter-software comparison, in terms of R

^{2}, RMSE, and normalized RMSE (nRMSE). Higher values of R

^{2}, lower values of RMSE, and lower values of nRMSE indicate higher accuracy.

## 3. Results

#### 3.1. PAI_{eff} and LAI_{eff} Estimation Results

_{eff-est}) were on average 55.8% of the PAI

_{true-ref}values, while the estimates of LAI (LAI

_{eff-est}) were on average 51.22% of LAI

_{true-ref}values. The comparison of the PAI

_{eff-est}and LAI

_{eff-est}among the three programs is shown in Figure 6. The PAI

_{eff}and LAI

_{eff}estimates from CIMES were slightly higher than those from Can_Eye were (as shown in Figure 6a,d. Compared to CIMES, the PAI

_{eff}and LAI

_{eff}estimates from Hemisfer were closer to Can_Eye (RMSD = 0.11 < 0.19 for PAI

_{eff-est}, and RMSD = 0.09 < 0.14 for LAI

_{eff-est}, as shown in Figure 6b,e. In general, the results of effective PAI and effective LAI from the three programs were consistent (R

^{2}≥ 0.8, RMSD ≤ 0.19).

#### 3.2. Comparison of PAI_{true} Estimation Accuracy

_{eff-est}from Can_Eye divided by the plot PAI

_{true-ref}, we obtained the clumping index values ($\mathsf{\lambda}$) of each forest plot, as shown in Table 5. This quantified the level of the clumping effect in each stand and assisted in the evaluation of clumping correction methods using different algorithms.

_{true-est}using different algorithms in Can_Eye, CIMES, and Hemisfer are presented in Figure 7, Figure 8 and Figure 9, respectively. The symbol of each plot in each figure was colored based on the clumping index ($\mathsf{\lambda}$) value according to the results in Table 5, and the size was adjusted according to the average leaf inclination angle of each plot in Table 1.

_{true-est}with PAI

_{true-ref}, the four algorithms in Can_Eye produced different PAI

_{true-est}values, with nRMSE ranging in (13.64%, 46.24%) (see Figure 7). The most accurate algorithm was the P57 algorithm, which used the gap fraction (at 57.5°) inversion combined with the LX clumping correction method (R

^{2}= 0.86, RMSE = 0.49, nRMSE = 13.64%). The least accurate algorithm in Can_Eye was Miller’s formula using the gap fraction at 0°~60° combined with the LX clumping correction method (RMSE = 1.68, nRMSE = 46.24%, see Figure 7).

_{true-est}values, with nRMSE ranging in (19.3%, 54.37%) (see Figure 8). The most accurate algorithm of CIMES was from the multiple direction gap fraction inversion at 0°~60° using the Campbell approach combined with the LX clumping correction (CAM_LX algorithm, R

^{2}= 0.73, RMSE = 0.7, nRMSE = 19.3%), while the least accurate algorithm was the Miller_CC57 method, with the nRMSE reaching 54.37% (see Figure 8).

_{true-est}values, with nRMSE ranging in (30.46%, 43.34%) (see Figure 9). The most accurate PAI

_{true-est}result from Hemisfer was obtained with the LX_Miller method (R

^{2}= 0.32, RMSE = 1.11, nRMSE = 30.46%), while the least accurate algorithm was the CC_2000 method, with the nRMSE reaching 43.34% (see Figure 9).

_{true}was the P57 method in Can_Eye, which used the gap fraction (at 57.5°) inversion combined with the LX clumping correction. Of note, there was a strong systematic underestimation of PAI

_{true}by Hemisfer, either for canopies with high or low clumping. The PAI

_{true}estimates from Hemisfer reached saturation at PAI values around three (Figure 9j). In comparison, the P57 method in Can_Eye could accurately correct the clumping effect until the vegetation reached high clumping (when λ < 0.45); then, it began to underestimate PAI

_{true}in forests with PAI above 4.5 (Figure 7d). There was neither a systematic underestimation nor overestimation of PAI

_{true}using the CAM_LX algorithm from CIMES (Figure 8a). In Figure 7d, Figure 8a and Figure 9j, there was no clear spatial distribution pattern of symbol sizes, implying that the average leaf inclination angle of each stand had little effect on the estimation of PAI

_{true}.

#### 3.3. Comparison of LAI_{true} Estimation Accuracy

_{true-est}, similar to the case of PAI

_{true-est}, different algorithms produced quite different LAI

_{true-est}results (nRMSE ranged in (15.7%, 53.24%) for Can_Eye, (24.21%, 70.64%) for CIMES, and (32.81%, 49.49%) for Hemisfer). Within each software program, the most accurate algorithm to estimate LAI

_{true}was the same as PAI

_{true}. For more details, the reader can refer to Figures S1–S3 in the Supplementary. In the following, only the most accurate algorithm in each software program was listed for an intercomparison.

_{true}compared to CIMES and Hemisfer (R

^{2}= 0.88 > 0.72, 0.49; RMSE = 0.45 < 0.7, 0.94; nRMSE = 15.7% < 24.21%, 32.81%, see Figure 10). There was a more severe underestimation of LAI

_{true}from the LX_Miller algorithm in Hemisfer than from CIMES and Can_Eye, even in forests with a moderate amount of leaves at the LAI value around 2.5 (Figure 10c). In comparison, the P57 method in Can_Eye started to underestimate LAI in forests with dense leaves, with an LAI value around four (Figure 10a). There was neither a systematic underestimation nor overestimation of LAI

_{true}from the CAM_LX algorithm in CIMES (Figure 10b).

## 4. Discussion

_{eff}estimates, but significantly different estimates of PAI

_{true}between Can_Eye and CIMES [37]. However, this previous study by Hall et al. only reported the inconsistency between Can_Eye and CIMES. The contribution of our research is that we proved that Can_Eye estimates more accurate PAI

_{true}and LAI

_{true}values than CIMES and Hemisfer. In addition, we identified the most accurate algorithm among all 37 algorithms as the P57 algorithm in Can_Eye. This can guide future users during the algorithm and software selection step when measuring in situ LAI from DHP. In addition, the use of virtual forests was not subject to inherent field measurement errors compared to litterfall collection or destructive sampling. It, thus, provides strong confidence in the results of the algorithm assessment.

_{true}and LAI

_{true}among different software programs (i.e., Can_Eye, CIMES, and Hemisfer) was due to different estimates of the clumping index. Many researchers have pointed out that the estimates of the clumping index remained a large source of uncertainty for LAI estimation [44,75]. There was a systematic underestimation of PAI

_{true}and LAI

_{true}using Hemisfer, even in forests with a moderate amount of leaves and low vegetation clumping (as shown in Figure 9g–l and Figure S3g–l). This may have been caused by the pure canopy segments with no gaps. In such cases, the underlying assumption “the segment contains gaps” in the LX method did not hold. Both the Can_Eye and CIMES software utilized a saturated LAI (${L}_{\mathrm{sat}}$) value to address this problem. In this experiment, ${L}_{\mathrm{sat}}$ was set as 10 for both Can_Eye and CIMES. Dissimilar to Can_Eye and CIMES, Hemisfer did not provide this ${L}_{\mathrm{sat}}$ solution and offered no details about how pure canopy segments were handled. This may be the reason for the systematic underestimation of PAI

_{true}and LAI

_{true}in Hemisfer.

_{true}and less accurate LAI

_{true}than Can_Eye (i.e., comparing Figure 8a with Figure 7b,c and comparing Figure S1a with Figure S2b,c in the Supplementary). This may be caused by the different inversion schemes implemented in Can_Eye and CIMES. Can_Eye introduces a regularization term in the cost function, which imposes constraints to improving the PAI estimates (see Equations (8) and (9)), while CIMES is not similarly constrained [4,19,59]. The higher accuracy of the v6.1 method than the v5.1 method in Can_Eye found in our study (R

^{2}= 0.86 > 0.81, RMSE = 0.5 < 0.58, nRMSE = 13.66% < 15.93%, as shown in Figure 7) corroborates the prediction of higher efficiency of the PAI regularization term when compared with the ALA regularization term [19].

_{true}and LAI

_{true}using CIMES and Hemisfer (as shown in Figure 8f,g, Figure 9a–f,m–r, Figures S2f,g and S3a–f,m–r in the Supplementary). This is in agreement with previous studies, which also found that the CC method underestimated the clumping effect (i.e., overestimation of λ [32,76].

_{true}and LAI

_{true}.

_{eff}from PAI

_{eff}correlated well with LAI

_{eff}when using a virtual forest [79]. The results of our study further illustrated that the subtraction of WAI

_{true}from PAI

_{true}correlated well with pre-defined LAI

_{true}values (Figure 10). This demonstrated the effectiveness of using leaf-on and leaf-off DHP to correct the woody effect and estimate LAI for broadleaf forests. This can also provide a reference for dynamic LAI

_{true}monitoring studies using repeated DHP for broader environmental applications [80].

## 5. Conclusions

^{2}> 0.8, RMSD < 0.2). However, after correcting for the clumping effect caused by a non-random vegetation distribution, a large inconsistency occurred between the estimates of true PAI and true LAI from different algorithms. The results demonstrated that Can_Eye more accurately estimated true LAI than CIMES and Hemisfer (with R

^{2}= 0.88 > 0.72, 0.49; RMSE = 0.45 < 0.7, 0.94; nRMSE = 15.7% < 24.21%, 32.81%). It was also found that the P57 algorithm, which used the 57.5° gap fraction inversion combined with the finite-length averaging (LX) clumping correction method implemented in Can_Eye, was the most accurate. This study highlighted the in situ LAI measurement uncertainty due to the choice of the LAI algorithm and demonstrated the effectiveness of using leaf-on and leaf-off to correct for the woody component’s effect on LAI estimation. From the results of this study, we recommend using the 57.5° gap fraction inversion combined with the finite-length averaging (LX) clumping correction method implemented in Can_Eye for LAI estimation in broadleaf forests. Further studies exploring coniferous and mixed forests are suggested to consolidate the protocol of in situ LAI measurements to reduce in situ LAI uncertainty.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The flowchart of accuracy comparison of different forest LAI estimation algorithms in digital hemispherical photography programs (including Can_Eye, CIMES, and Hemisfer).

**Figure 2.**Example of the 3D individual tree models for virtual forests construction: (

**a**) a 10 m beech tree without leaves, (

**b**) a 10 m beech tree with leaves, (

**c**) a 30 m oak tree without leaves, and (

**d**) a 30 m oak tree with leaves.

**Figure 3.**The distribution of trees with different heights in all 30 virtual forest stands (H5, H10, H15, H20, H25, H30 refer to trees of 5, 10, 15, 20, 25, and 30 m heights, respectively).

**Figure 5.**Synthetic digital hemispherical photography (DHP) of (

**a**) Plot2 in leaf-on condition, (

**b**) Plot2 in leaf-off condition, (

**c**) Plot30 in leaf-on condition, and (

**d**) Plot30 in leaf-off condition.

**Figure 6.**Correlation of the effective plant area index estimates (PAI

_{eff-est}) and effective leaf area index estimates (LAI

_{eff-est}) using three digital hemispherical photography programs (on average, PAI

_{eff-est}were 55.8% of the PAI

_{true-ref}values, while LAI

_{eff-est}were 51.22% of LAI

_{true-ref}values).

**Figure 7.**${\mathrm{PAI}}_{\mathrm{true}}$ results from Can_Eye using different algorithms including the (

**a**) Miller (

**b**) v5.1 (

**c**) v6.1, and (

**d**) P57 algorithm; the best result was produced by the Can_Eye P57 algorithm. A smaller symbol indicates a smaller average leaf inclination angle (ALA), while a larger symbol indicates a higher ALA.

**Figure 8.**${\mathrm{PAI}}_{\mathrm{true}}$ results from CIMES using different algorithms including the (

**a**) CAM_LX, (

**b**) CMP_WT, (

**c**) LOGCAM, (

**d**) LANG_LX, (

**e**) MLR, (

**f**) Miller_CC57, (

**g**) Miller_CC, (

**h**) Miller_CLX57, and (

**i**) Miller_CLX algorithm; the best result was produced by the CIMES CAM_LX algorithm. A smaller symbol indicates a smaller average leaf inclination angle (ALA), while a larger symbol indicates a higher ALA; the Miller_CLX method only produced estimates for 26 out of 30 plots with results.

**Figure 9.**${\mathrm{PAI}}_{\mathrm{true}}$ results from Hemisfer using different algorithms including the (

**a**) CC_2000, (

**b**) CC_Gonsamo, (

**c**) CC_Lang, (

**d**) CC_Miller, (

**e**) CC_NC, (

**f**) CC_Thimonier, (

**g**) LX_2000, (

**h**) LX_Gonsamo, (

**i**) LX_Lang, (

**j**) LX_Miller, (

**k**) LX_NC, (

**l**) LX_Thimonier, (

**m**) SCC_2000, (

**n**) SCC_Gonsamo, (

**o**) SCC_Lang, (

**p**) SCC_Miller, (

**q**) SCC_NC, (

**r**) SCC_Thimonier, (

**s**) WT_2000, (

**t**) WT_Gonsamo, (

**u**) WT_Lang, (

**v**) WT_Miller, (

**w**) WT_NC, (

**x**) WT_Thimonier algorithm_; the best result was produced by the Hemisfer LX_Miller algorithm. A smaller symbol indicating a smaller average leaf inclination angle (ALA) while a larger symbol indicating a higher ALA.

**Figure 10.**Accuracy of the true leaf area index estimates (LAI

_{true-est}, calculated from PAI

_{true-est}minus WAI

_{true-est}) using three digital hemispherical photography programs including (

**a**) Can_Eye, (

**b**) CIMES, and (

**c**) Hemisfer compared to ground reference values (LAI

_{true-ref}). A smaller symbol indicates a smaller average leaf inclination angle (ALA) while a larger symbol indicates a higher ALA.

Stand | Plot Name | Stand Size (m) | Plot Radius (m) | Stand PAI_{true-ref} ^{(1)} | Stand LAI_{true-ref} ^{(2)} | Plot PAI_{true-ref} ^{(3)} | Plot LAI_{true-ref} ^{(4)} | ALA^{(5)} (°) |
---|---|---|---|---|---|---|---|---|

F1 | Plot 1 | 120 × 120 | 25 | 1.43 | 0.52 | 1.39 | 0.49 | 5 |

F2 | Plot 2 | 120 × 120 | 25 | 1.48 | 0.89 | 1.30 | 0.79 | 30 |

F3 | Plot 3 | 120 × 120 | 25 | 1.77 | 1.13 | 1.57 | 0.99 | 68 |

F4 | Plot 4 | 120 × 120 | 25 | 2.13 | 1.22 | 2.03 | 1.13 | 78 |

F5 | Plot 5 | 120 × 120 | 25 | 2.21 | 1.38 | 2.15 | 1.35 | 8 |

F6 | Plot 6 | 120 × 120 | 25 | 2.73 | 1.90 | 2.19 | 1.54 | 32 |

F7 | Plot 7 | 120 × 120 | 25 | 2.90 | 2.11 | 2.28 | 1.64 | 65 |

F8 | Plot 8 | 120 × 120 | 25 | 2.34 | 1.58 | 2.48 | 1.66 | 28 |

F9 | Plot 9 | 120 × 120 | 25 | 2.44 | 1.70 | 2.66 | 1.82 | 75 |

F10 | Plot 10 | 120 × 120 | 25 | 3.18 | 2.33 | 3.00 | 2.18 | 10 |

F11 | Plot 11 | 120 × 120 | 25 | 3.02 | 2.28 | 2.98 | 2.26 | 53 |

F12 | Plot 12 | 120 × 120 | 25 | 3.67 | 2.81 | 3.32 | 2.54 | 35 |

F13 | Plot13 | 120 × 120 | 25 | 3.96 | 3.17 | 3.24 | 2.58 | 38 |

F14 | Plot14 | 120 × 120 | 25 | 4.06 | 3.27 | 3.29 | 2.61 | 50 |

F15 | Plot15 | 120 × 120 | 25 | 3.39 | 2.55 | 3.56 | 2.65 | 25 |

F16 | Plot16 | 120× 120 | 25 | 3.68 | 2.98 | 3.66 | 2.96 | 64 |

F17 | Plot17 | 120 × 120 | 25 | 4.46 | 3.47 | 4.04 | 3.12 | 12 |

F18 | Plot18 | 120 × 120 | 25 | 4.22 | 3.43 | 4.05 | 3.27 | 23 |

F19 | Plot19 | 120 × 120 | 25 | 4.53 | 3.82 | 4.01 | 3.39 | 60 |

F20 | Plot20 | 120 × 120 | 25 | 5.03 | 4.12 | 4.30 | 3.54 | 48 |

F21 | Plot21 | 120× 120 | 25 | 4.88 | 4.09 | 4.45 | 3.73 | 20 |

F22 | Plot22 | 120 × 120 | 25 | 5.46 | 4.58 | 4.54 | 3.79 | 45 |

F23 | Plot23 | 120 × 120 | 25 | 5.02 | 4.23 | 4.52 | 3.82 | 40 |

F24 | Plot24 | 120 × 120 | 25 | 6.15 | 5.29 | 4.70 | 4.07 | 42 |

F25 | Plot25 | 120 × 120 | 25 | 5.32 | 4.36 | 5.10 | 4.17 | 15 |

F26 | Plot26 | 120 × 120 | 25 | 6.38 | 5.53 | 5.10 | 4.44 | 72 |

F27 | Plot27 | 120 × 120 | 25 | 5.79 | 4.98 | 5.34 | 4.53 | 70 |

F28 | Plot28 | 120 × 120 | 25 | 5.83 | 4.92 | 5.51 | 4.63 | 18 |

F29 | Plot29 | 120 × 120 | 25 | 6.09 | 5.18 | 5.92 | 5.07 | 58 |

F30 | Plot30 | 120 × 120 | 25 | 6.14 | 5.42 | 6.15 | 5.39 | 55 |

^{(1)}Stand PAI

_{true-ref}: the reference value of true PAI in the stand (120 m×120 m) from 1.5 m above ground.

^{(2)}Stand LAI

_{true-ref}: the reference value of true LAI in the stand (120 m×120 m) from 1.5 m above ground.

^{(3)}Plot PAI

_{true-ref}: the reference value of true PAI in the circular plot (25 m radius) from 1.5 m above ground.

^{(4)}Plot LAI

_{true-ref}: the reference value of true LAI in the circular plot (25 m radius) from 1.5 m above ground.

^{(5)}ALA: average leaf inclination angle.

Algorithm Abbreviation | Basic Principle | References |
---|---|---|

P57 | 1. Use of the gap fraction at 57.5° (55°~60°) 2. The $G\left(\mathsf{\theta}\right)$ was approximated as 0.5 regardless of $g\left(\theta \right)$types 3. Clumping correction was based on the LX method 4. ${L}_{\mathrm{sat}}$ was set as 10 for pure segments with no gaps | [19,63] |

v5.1 | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. The $g\left(\theta \right)$ which determined $G\left(\theta \right)$was modeled by the ALA using the ellipsoidal distribution 3. PAI and ALA were inversed using a look-up table scheme, with the cost function constrained by a term of ALA (the retrieved ALA value must be close to 60° ± 30°) 4. Clumping correction was based on the LX method 5. ${L}_{\mathrm{sat}}$ was set as 10 for pure segments with no gaps | [19,60,63] |

v6.1 | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. The $g\left(\theta \right)$ was modeled by the ALA using the ellipsoidal distribution 3. PAI and ALA were inversed using a lookup table scheme, with the cost function constrained by a term of PAI ^{57} (the retrieved PAI value that must be close to the one retrieved from the annulus at 57.5°)4. Clumping correction was based on the LX method 5. ${L}_{\mathrm{sat}}$ was set as 10 for pure segments with no gaps | [19,60,63] |

Miller | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. Use of Miller’s formula to estimate PAI _{eff}3. Clumping correction based on the LX method 4. ${L}_{\mathrm{sat}}$ was set as 10 for pure segments with no gaps | [19,58] |

Algorithm Abbreviation | Basic Principle | References |
---|---|---|

CAM_LX | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. The $g\left(\theta \right)$ was modeled by the ALA using the ellipsoidal distribution 3. Clumping correction was based on the LX method 4. L _{sat} was set as 10 for pure segments with no gaps | [4,59,60,63] |

CMP_WT | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. The $g\left(\theta \right)$ was modeled by the ALA using the ellipsoidal distribution 3. Clumping correction was based on the WT method 4. L _{sat} was set as 10 for pure segments with no gaps | [4,59,60,67] |

LOGCAM | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. The $g\left(\theta \right)$ was modeled by the ALA using the ellipsoidal distribution 3. Clumping correction was based on a modified LX method using variable azimuthal segmentations of the hemisphere | [4,59,60,63] |

LANG_LX | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. Use of Lang’s regression method to estimate PAI _{eff}3. Clumping correction was based on the LX method 4. L _{sat} was set as 10 for pure segments with no gaps | [4,59,63,71] |

MLR | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. Use of Lang’s regression method to estimate PAI _{eff}3. Clumping correction was based on a modified LX method using variable azimuthal segmentations of the hemisphere | [4,59,71] |

Miller_CC57 | 1. Use of the gap fraction at 57.5° (55°~60°) 2. Use of Miller’s formula to estimate PAI _{eff}3. Clumping correction was based on the CC method | [4,58,59,64] |

Miller_CC | 1. Use of the gap fraction at 0°~60° 2. Use of Miller’s formula to estimate PAI _{eff}3. Clumping correction was based on the CC method | [4,58,59,64] |

Miller_CLX57 | 1. Use of the gap fraction at 57.5° (55°~60°) 2. Use of Miller’s formula to estimate PAI _{eff}3. Clumping correction was based on the CLX method | [4,58,59,66] |

Miller_CLX | 1. Use of the gap fraction at 0°~60° 2. Use of Miller’s formula to estimate PAI _{eff}3. Clumping correction was based on the CLX method | [4,58,59,66] |

AlgorithmAbbreviation | Basic Principle | References |
---|---|---|

CC_2000 | 1. Use of the gap fraction at 0°~60° 2. Use of the LI-COR LAI-2000 method to estimate PAI _{eff}3. Clumping correction was based on the CC method | [64,72,73] |

CC_Gonsamo | 1. Use of the gap fraction at 0°~60° 2. Use of the Lang Robust regression method proposed by Gonsamo to estimate PAI _{eff}3. Clumping correction was based on the CC method | [64,73,74] |

CC_Lang | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. Use of Lang’s regression method to estimate PAI _{eff}3. Clumping correction was based on the CC method | [64,71,73] |

CC_Miller | 1. Use of the gap fraction at 0°~60° 2. Use of Miller’s formula to estimate PAI _{eff}3. Clumping correction was based on the CC method | [58,73] |

CC_NC | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. The $g\left(\theta \right)$ was modeled by the ALA using the ellipsoidal distribution 3. Clumping correction was based on the CC method | [62,64,73] |

CC_Thimonier | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. The $g\left(\theta \right)$ was modeled by the ALA using the weighted ellipsoidal distribution 3. Clumping correction was based on the CC method | [30,73] |

LX_2000 | 1. Use of the gap fraction at 0°~60° 2. Use of the LI-COR LAI- 2000 method to estimate PAI _{eff}3. Clumping correction was based on the LX method | [63,72,73] |

LX_Gonsamo | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. Use of the Lang Robust regression method proposed by Gonsamo to estimate PAI _{eff}3. Clumping correction was based on the LX method | [63,73,74] |

LX_Lang | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. Use of Lang’s regression method to estimate PAI _{eff}3. Clumping correction was based on the LX method | [63,71,73] |

LX_Miller | 1. Use of the gap fraction at 0°~60° 2. Use of Miller’s formula to estimate PAI _{eff}3. Clumping correction was based on the LX method | [58,63,73] |

LX_NC | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. The $g\left(\theta \right)$ was modeled by the ALA using the ellipsoidal distribution 3. Clumping correction was based on the LX method | [62,63,73] |

LX_Thimonier | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. The $g\left(\theta \right)$ was modeled by the ALA using the weighted ellipsoidal distribution 3. Clumping correction was based on the LX method | [30,63,73] |

SCC_2000 | 1. Use of the gap fraction at 0°~60° 2. Use of the LI-COR LAI- 2000 method to estimate PAI _{eff}3. Use of Schleppi’s approach to correct for within annulus non-linearity of path lengths4.Clumping correction was based on the CC method | [20,64,72,73] |

SCC_Gonsamo | 1. Use of the gap fraction at 0°~60° 2. Use of the Lang Robust regression method proposed by Gonsamo to estimate PAI _{eff}3. Use of Schleppi’s approach to correct for within annulus non-linearity of path lengths 4. Clumping correction was based on the CC method | [20,64,73,74] |

SCC_Lang | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. Use of Lang’s regression method to estimate PAI _{eff}3. Use of Schleppi’s approach to correct for within annulus non-linearity of path lengths4. Clumping correction was based on the CC method | [20,71,73] |

SCC_Miller | 1. Use of the gap fraction at 0°~60° 2. Use of Miller’s formula to estimate PAI _{eff}3. Use of Schleppi’s approach to correct for within annulus non-linearity of path lengths 4. Clumping correction was based on the CC method | [20,58,64,73] |

SCC_NC | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. The $g\left(\mathsf{\theta}\right)$ was modeled by the ALA using the ellipsoidal distribution 3. Use of Schleppi’s approach to correct for within annulus non-linearity of path lengths 4. Clumping correction was based on the CC method | [20,62,64,73] |

SCC_Thimonier | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. The $g\left(\mathsf{\theta}\right)$ was modeled by the ALA using the weighted ellipsoidal distribution 3. Use of Schleppi’s approach to correct for within annulus non-linearity of path lengths 4. Clumping correction was based on the CC method | [20,30,64,73] |

WT_2000 | 1. Use of the gap fraction at 0°~60° 2. Use of the LI-COR LAI-2000 method to estimate PAI _{eff}3. Clumping correction was based on the WT method | [67,72,73] |

WT_Gonsamo | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. Use of the Lang Robust regression method proposed by Gonsamo to estimate PAI _{eff}3. Clumping correction was based on the WT method | [67,73,74] |

WT_Lang | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. Use of Lang’s regression method to estimate PAI _{eff}3. Clumping correction was based on the WT method | [67,71,73] |

WT_Miller | 1. Use of the gap fraction at 0°~60° 2. Use of Miller’s formula to estimate PAI _{eff}3. Clumping correction was based on the WT method | [58,67,73] |

WT_NC | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. The $g\left(\theta \right)$ was modeled by the ALA using the ellipsoidal distribution 3. Clumping correction was based on the WT method | [62,67,73] |

WT_Thimonier | 1. Use of the gap fraction at 0$\xb0~60\xb0$ 2. The $g\left(\theta \right)$ was modeled by the ALA using the weighted ellipsoidal distribution 3. Clumping correction was based on the WT method | [30,67,73] |

Plot Name | $\mathit{\lambda}$ | Plot Name | $\mathit{\lambda}$ | Plot Name | $\mathit{\lambda}$ |
---|---|---|---|---|---|

Plot1 | 0.52 | Plot 11 | 0.63 | Plot 21 | 0.46 |

Plot 2 | 0.60 | Plot 12 | 0.54 | Plot 22 | 0.49 |

Plot 3 | 0.78 | Plot 13 | 0.56 | Plot 23 | 0.42 |

Plot 4 | 0.67 | Plot 14 | 0.56 | Plot 24 | 0.46 |

Plot 5 | 0.55 | Plot 15 | 0.52 | Plot 25 | 0.43 |

Plot 6 | 0.72 | Plot 16 | 0.57 | Plot 26 | 0.46 |

Plot 7 | 0.75 | Plot 17 | 0.49 | Plot 27 | 0.42 |

Plot 8 | 0.54 | Plot 18 | 0.49 | Plot 28 | 0.38 |

Plot 9 | 0.68 | Plot 19 | 0.53 | Plot 29 | 0.42 |

Plot 10 | 0.54 | Plot 20 | 0.48 | Plot 30 | 0.39 |

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## Share and Cite

**MDPI and ACS Style**

Liu, J.; Li, L.; Akerblom, M.; Wang, T.; Skidmore, A.; Zhu, X.; Heurich, M.
Comparative Evaluation of Algorithms for Leaf Area Index Estimation from Digital Hemispherical Photography through Virtual Forests. *Remote Sens.* **2021**, *13*, 3325.
https://doi.org/10.3390/rs13163325

**AMA Style**

Liu J, Li L, Akerblom M, Wang T, Skidmore A, Zhu X, Heurich M.
Comparative Evaluation of Algorithms for Leaf Area Index Estimation from Digital Hemispherical Photography through Virtual Forests. *Remote Sensing*. 2021; 13(16):3325.
https://doi.org/10.3390/rs13163325

**Chicago/Turabian Style**

Liu, Jing, Longhui Li, Markku Akerblom, Tiejun Wang, Andrew Skidmore, Xi Zhu, and Marco Heurich.
2021. "Comparative Evaluation of Algorithms for Leaf Area Index Estimation from Digital Hemispherical Photography through Virtual Forests" *Remote Sensing* 13, no. 16: 3325.
https://doi.org/10.3390/rs13163325