Sparse tensor networks for individual tree diameter estimation in mixed forests

Clara Lößl; Fabian Ewald Fassnacht; Viacheslav Barkov

cs.CVOct 27, 2026

Sparse tensor networks for individual tree diameter estimation in mixed forests

Authors: Clara Lößl, Fabian Ewald Fassnacht, Viacheslav Barkov

Organizations: Joint Lab Artificial Intelligence and Data Science, Osnabrück University, Osnabrück, Germany · Department of Agromechatronics, Leibniz Institute for Agricultural Engineering and Bioeconomy (ATB), Potsdam, Germany · Chair of Agromechatronics, Technische Universität Berlin, Berlin, Germany · Institute of Geomatics, University of Natural Resources and Life Sciences (BOKU), Vienna, Austria

Abstract

Measurements of tree diameter at breast height (DBH) are fundamental to forest inventories, but manual surveys across entire stands or regions are impractical. UAV-borne laser scanning (ULS) enables scalable automated forest surveys, yet canopy occlusion makes accurate DBH estimation from ULS point clouds non-trivial. Limited stem-level point density of ULS data constrains approaches relying on detailed stem reconstruction, such as Quantitative Structure Models (QSMs) and circle fitting. Empirical machine learning methods based on predefined point cloud metrics are an alternative. New deep learning approaches offer an even more principled solution, allowing models to learn directly from point clouds. Sparse tensor networks have shown particular promise, transforming forestry tasks such as tree species classification. We investigate whether these strengths can be extended to estimate individual tree DBH from ULS data across four mixed-species temperate forest plots in Germany. We further advance existing sparse tensor networks by introducing TreeNext, a lightweight deep learning architecture for direct individual tree parameter estimation from voxelized LiDAR point clouds. We present a fully automated forest inventory pipeline from individual tree instance segmentation to DBH regression, and quantify the impact of segmentation errors by comparing performance on automatically segmented and ground truth instances. TreeNext achieves an $R^2$ of 0.8355 on automatically segmented trees and 0.9380 on ground truth segments, outperforming strong tabular baselines including Random Forest and XGBoost. Segmentation quality emerges as the dominant error source in the automated pipeline. TreeNext proves highly data-efficient, performing well even in small data scenarios. Our findings establish sparse tensor networks as a practical foundation for automated individual tree DBH estimation from ULS.

Figures & tables

Figure 1: Geographic location of the study plots. (a) Location of Karlsruhe (KA) and Bretten (BR) within Baden-Württemberg, Germany. (b) Map of the Karlsruhe plot (KA09). (c) Maps of the three Bretten plots (BR01, BR03, BR05). The plots are 1-ha forest stands used throughout the study. Base map copyright: OpenStreetMap contributors.

Figure 2: Distribution of diameter at breast height (DBH) values for all reference trees across all study sites. Vertical dashed lines mark the mean, median, minimum, and maximum DBH as indicated in the legend.

Figure 3: Visual comparison of ground truth and automatically segmented tree instances. (a,c,e) Ground truth tree instances. (b,d,f) Automatic instances from SegmentAnyTree using default inference settings. Colors indicate distinct tree instances.

Figure 4: Illustration of voxelized tree representation. (a) Raw individual tree point cloud. (b) Fine voxelization (grid size 0.05 m). (c) Coarse voxelization (grid size 0.2 m).

Figure 5: TreeNext network architecture. Individual tree point clouds are voxelized while preserving LiDAR return attributes as input channels. The model uses depthwise separable sparse convolutions with Global Response Normalization (GRN), hierarchical downsampling, and concatenated global max/average pooling.

Figure 6: Distribution of reference tree DBH values by matching status after segmentation-based instance assignment. Matched trees (green) were successfully linked to a predicted segmented instance. Unmatched trees (red) had no corresponding instance. All unmatched trees have a DBH

<

25 cm.

	Automatically segmented trees		Ground truth segments		Parameters
Model	Mean $R^{2}$	Ensemble $R^{2}$	Mean $R^{2}$	Ensemble $R^{2}$
Sparse Tensor Networks
TreeNext (Proposed)	$\mathbf{0.8266\pm 0.0018}$	$\mathbf{0.8355}$	$\mathbf{0.9322\pm 0.0027}$	$\mathbf{0.9380}$	5.19 M
MSENet50	$0.7872\pm 0.0069$	$0.8004$	$0.9076\pm 0.0045$	$0.9157$	48.71 M
MSENet14	$0.7893\pm 0.0025$	$0.7963$	$0.8944\pm 0.0028$	$0.8996$	14.40 M
Tabular Regression with Point Cloud Metrics

Table 1: Model performance comparison. The results are calculated after 5-fold cross validation. Mean and ensemble

R^{2}

are calculated over repeated cross validation runs (k=5). Model parameter counts for sparse tensor networks are reported in millions (M).

Figure 7: Scatter plots of predicted versus reference DBH for TreeNext and the tabular regression baselines on automatically segmented trees. Points are colored by broadleaf and conifer species group. The dashed line indicates the 1:1 relationship.

Figure 8: Heatmap of TreeNext DBH estimation error (rRMSE, %) as a function of segmentation IoU (across 10 uniform bins) and ground truth DBH (across 10 quantile bins). Bins with at least ten trees are shown.

Figure 9: Illustration of automatic segmentation issues of two selected trees with high DBH estimation error. The scatter plot shows predicted versus reference DBH for TreeNext on automatically segmented trees. Points are colored by broadleaf and conifer species group. Two outliers are highlighted: (a) a Larix decidua conifer tree, and (b) a Fagus sylvatica broadleaf tree. For each outlier, the automatically segmented point cloud and the corresponding ground truth segment are shown.

Figure 10: Effect of training set size on performance for TreeNext, its post-hoc ensemble (TreeNext Ensemble), and the strongest tabular baseline (XGBoost) across fractions of the available training data. Sample counts correspond to one outer training fold in cross validation.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 11: Effect of post-hoc ensemble size on TreeNext. (a) Relative improvement compared to a single model (k=1). (b) Marginal gain when increasing k. Shaded bands show

\pm 1

SD across repeated runs. The vertical dashed line marks k=5 used in the main results.

Sparse Tensor Networks
	Automatically segmented trees		Ground truth segments
Model	RMSE	rRMSE (%)	RMSE	rRMSE (%)
TreeNext (Proposed)	$\mathbf{7.3246\pm 0.0388}$	$\mathbf{23.72\pm 0.13}$	$\mathbf{4.6074\pm 0.0915}$	$\mathbf{15.60\pm 0.31}$
MSENet50	$8.1143\pm 0.1325$	$26.27\pm 0.43$	$5.3767\pm 0.1308$	$18.21\pm 0.44$
MSENet14	$8.0751\pm 0.0485$	$26.15\pm 0.16$	$5.7497\pm 0.0769$	$19.47\pm 0.26$
Tabular Regression with Point Cloud Metrics

Appendix

Table 2: Model performance comparison. The results are calculated after 5-fold cross validation. Mean RMSE and relative RMSE (rRMSE) are calculated over repeated cross validation runs (k=5).

Category	#	Definition
Basic height stats	9	Number of points, $z_{\min}$ , $z_{\max}$ , mean, variance, SD, $z_{cv}=100\cdot\text{SD}/\text{mean}$ , skewness, and kurtosis computed from all return heights.
Percentiles	21	Height percentiles: 1st; 5th–95th (step 5); 99th.
Canopy density (cumulative)	9	Cumulative percentage of returns in 10 equal-width height bins covering the range $[z_{\min},z_{\max}]$ .
Dispersion	5	IQR; mean and median absolute deviations; canopy relief ratio $(\text{mean}-z_{\min})/(z_{\max}-z_{\min})$ ; Shannon entropy of $z>0$ using 1 m vertical layers.
Echo-based (aggregate)	13	Counts and percentages of first, intermediate, last, single, and multiple returns; ratios of last-to-first, intermediate-to-first, and multiple-to-single returns.
Echo-based (per return order)	var.	Point count $n_{r}$ for each return order $r$ present in the cloud ( $r=1,2,\dots$ ).

Appendix

Table 3: List of LiDAR point-cloud metrics extracted for each individual tree and used as input features for the tabular baseline models.

Algorithm	Parameter	Search grid
XGBoost	learning_rate	$[0.001,\,0.3]_{\log}$
	max_depth	$\{3,\dots,10\}$
	min_child_weight	$\{1,\dots,10\}$
	subsample	$[0.5,\,1.0]$
	colsample_bytree	$[0.5,\,1.0]$
	gamma	$[0,\,5]$

Appendix

Table 4: Hyperparameter grids for classical baselines. Bracketed intervals were sampled uniformly; ”log” indicates log-uniform sampling.

Tensor Networks

Tensor networks represent high-dimensional data and quantum states as interconnected low-rank tensors to enable efficient computation, learning, and simulation.

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Remote Sensing

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Accurate tree-level forest monitoring using laser scanning data requires reliable tree delineation, consistent tree correspondence across multitemporal point clouds, and accurate estimation of tree attributes and their change. Reconstructing tree growth in boreal forests is challenging due to the scarcity of historical stem-level data, propagation of errors from older sensors into change estimation, and growth rates with a magnitude of measurement uncertainty. This study investigates a framework for estimating individual tree diameter at breast height (DBH) and stem volume growth using 136 point clouds acquired between 2014--2025 with 11 scanners on airborne (ALS), mobile (MLS), and terrestrial laser scanning (TLS) platforms across boreal forest test sites. Trees were delineated from an MLS point cloud using deep learning-based segmentation which was transferred to the remaining point clouds, resulting in reliable multitemporal tree correspondence. Stem curves were derived from MLS/TLS data, with ALS data used for height estimation, enabling DBH and volume estimation and time series. A height growth-based scaling model was used to reconstruct stem attributes across time and estimate growth. Results showed that modeled growth achieved higher agreement with manual growth estimates than differencing independently estimated attributes from point clouds. The modeled-manual 5- and 10-year growth RMSEs were 55--111% and 26--67% for DBH, and 31--87% and 21--67% for volume, respectively, depending on plot difficulty. The scaling model was temporally robust, with errors remaining stable or stabilizing after 5--6 years, reaching maximum RMSEs of 8--12% for DBH and 12--23% for volume after 12 years. Combining MLS/TLS-derived stem measurements with multitemporal ALS-derived heights provided a robust framework for individual tree growth estimation without requiring multiple under-canopy scans.

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