TreeDGS: Dynamic Graphs & Aerial DBH Sensing

• TreeDGS is a dual-framework approach that combines a dynamic compressed tree-depth decomposition for efficient MSO₁ property testing with a 3D Gaussian Splatting pipeline for remote forest measurement.
• The aerial DBH estimation pipeline leverages continuous differentiable scene representations, optimized SfM–MVS initialization, and opacity‐weighted RANSAC circle fitting to achieve a 39% RMSE reduction compared to LiDAR methods.
• The dynamic data structure framework maintains compressed multicabinet representations that support rapid vertex/edge updates and constant-time MSO₁ queries, ensuring structural consistency in bounded tree-depth graphs.

TreeDGS encompasses two distinct technical contexts: (1) a dynamic compressed tree-depth decomposition data structure for graphs with applications to MSO₁ property testing, and (2) a pipeline for aerial forest measurement using 3D Gaussian Splatting (3DGS) for direct tree diameter at breast height (DBH) estimation. Both share the label "TreeDGS" in the literature but represent independent methodological frameworks. This entry documents both developments, clarifying scope and operational principles in each.

1. Three-Dimensional Gaussian Splatting for Distant DBH Measurement

TreeDGS, as introduced in the context of remote sensing, denotes a pipeline for reconstructing and measuring vertical stem geometry—specifically, DBH—from posed aerial imagery (Shaheen et al., 19 Jan 2026). The method leverages continuous differentiable scene representations via 3D Gaussian Splatting (3DGS), facilitating accurate, densifiable reconstruction despite large acquisition distances and the sparse visual presence of stems. The pipeline is optimized for direct, object-level estimation in natural, complex forest scenes where traditional photogrammetric and LiDAR approaches are less effective.

1.1 Scene Representation and Optimization

The pipeline models the forest scene using a set of $K$ anisotropic Gaussians:

$$G_k = (\mu_k, s_k, R_k, \alpha_k, c_k),\quad k=1\ldots K$$

where $\mu_k \in \mathbb{R}^3$ (center), $s_k \in \mathbb{R}_+^3$ (scale), $R_k \in SO(3)$ (orientation), $\alpha_k \in [0,1]$ (opacity), and $c_k$ (appearance parameterization).

Per-view rendering employs depth-sorted, front-to-back compositing with local alpha-rasterization, producing

$$\hat I_i(u) = \sum_{k=1}^K w_k(u) c_k(u), \quad w_k(u)=\alpha_{k,i}(u) T_{k-1}(u)$$

optimized using photometric loss plus regularizers on scale, opacity, and color. The RaDe-GS approach provides local surface rasterization for fast, depth-aware queries and adaptively modulates hyper-parameters via reinforcement learning of Gaussian Splatting (RLGS).

1.2 Structure-from-Motion and Initialization

Initialization commences with Structure-from-Motion (SfM, e.g. TopicFM+ matcher and GLOMAP bundle adjustment) to recover camera matrices ($P_i = K_i [R_i|t_i]$) and a sparse scene point cloud ($X_{\rm sparse}$). Multi-View Stereo (OpenMVS) densifies the reconstruction to $X_{\rm dense}$, which seeds the Gaussian field. Initial Gaussian parameters are set via sampling, fixed spread ($\sim$0.1 m), identity orientation, low opacity, and appearance averaging.

1.3 RaDe-GS Surface Sampling and Reliability Scoring

A two-stage process extracts a dense, surface-consistent cloud:

• Candidate sampling: For each splat, $M$ random 3D offsets are generated and mapped, each retained with probability $\alpha_i$.
• Depth-aware filtering: For each sample $p$, per-view cumulative opacity $\tilde\alpha_v(p)$ is accumulated by compositing $\alpha_{k,v}(p)$, where opacity is dynamically clamped behind local depth planes.
• Multi-view scoring: Each point $p$ is associated with $\bar\alpha(p) = \min_{v\in V(p)} \tilde\alpha_v(p)$ and $w(p)=|V(p)|$, reflecting its reliability across views.

1.4 Trunk Segmentation and Opacity-Weighted Circle Fitting

Segmented tree trunks (via ForestFormer3D) are sliced horizontally near breast height. Slice points, weighted by $\bar\alpha_k$, inform a RANSAC-driven fit of the solid-circle model

$$\min_{x_0,y_0,r} \sum_{k=1}^n w_k \left( \|q_k-(x_0,y_0)\|^2 - r^2 \right)^2$$

through opacity-weighted hypothesis sampling and inlier scoring. Taper profiles are regressed as $\hat d(h) = \beta_0 + \beta_1 h$, with DBH extracted at $h_\mathrm{BH}=1.37$–$1.40$ m.

1.5 Empirical Performance

Measurement evaluation on 10 circular plots (210 field-measured trees, DBH 7.9–55.7 cm) yields RMSE of 4.79 cm (MAE 3.70 cm, ME –0.38 cm, 189/190 success) for TreeDGS versus 7.91 cm RMSE (MAE 5.04 cm, ME 1.64 cm, 187/190 success) for UAV-LiDAR cylinder fitting. This reflects a 39% RMSE reduction for TreeDGS using RGB-only inputs (Shaheen et al., 19 Jan 2026).

2. Dynamic Data Structure for Tree-Depth Decomposition

The alternate meaning of TreeDGS refers to a dynamic data structure for maintaining a compressed tree-depth decomposition ("tree-depth graph structure") of a graph $G$ with bounded tree-depth $D$, supporting dynamic insertion and deletion of vertices and edges while preserving $td(G) \leq D$ (Dvorak et al., 2013). The design explicitly accommodates dynamic MSO₁-property testing.

2.1 Tree-Depth Decomposition Framework

• Closure and Minimum Height: The tree-depth $td(G)$ is the smallest $t$ such that $G$ is a subgraph of the closure of a rooted forest $F$ of height $t$; closure connects each ancestor–descendant pair.
• Compressed Representation: The explicit data structure encodes a decomposition with each node labeled by adjacency and a multiplicity vector (capturing counts of isomorphic subtrees up to a threshold $S$).
• Equivalence and Compression: Isomorphic limbs (subtrees) are merged into "cabinets"; their original nodes form "drawers". Only one representative per isomorphism type is retained, with multiplicity capped at $S$.

2.2 Supported Operations and Complexity

The following dynamic operations are supported (for fixed $D$ and MSO formula $\varphi$):

Operation	Time Complexity
Insert isolated $v$	$O(1)$
Delete isolated $v$	$O(1)$
Delete edge $e$	$O(D)$
Insert edge $e$	$O(D \cdot (D + l_D))$
MSO-property query	$O(1)$

$l_D$ is the function expressing the (tower-exponential) number of minimal tree types as a function of $D$ and $|\varphi|$.

2.3 Algorithmic Maintenance

• Edge Deletion: Extract the path from root to endpoint, update adjacency bits, split on disconnection, and merge isomorphic cabinets. Per-update work is $O(D)$.
• Edge Insertion: If endpoints are already ancestor–descendant, dual to deletion. Otherwise, iteratively reroot the decomposition, searching for a new root using a precomputed table of minimal tree types, followed by path extraction and merging. Worst-case time $O(D \cdot (D + l_D))$.
• Path Extraction: Trace from target cabinet to root, splitting drawers when multiplicity exceeds one.
• Cleaning Cabinets: When subtrees become isomorphic, merge cabinets; splitting occurs only during path traversal or rerooting.

2.4 Constant-Time MSO₁ Property Testing

The root cabinet stores a Boolean indicating whether the compressed decompression satisfies a fixed MSO₁ formula $\varphi$. On update, this flag is recomputed using a precomputed table (size $l_D$), so that $\varphi$-queries are $O(1)$. Updates propagate through $O(D)$ cabinets, each handled in $O(1)$.

2.5 Preprocessing and Correctness Guarantees

• Initialization: Any initial tree-depth decomposition is found in $O(n)$, compressed bottom-up in $O(n)$, and the minimal-type table of size $l_D$ is constructed in time $2^{O(D)}\,\text{poly}(|\varphi|)$.
• Correctness: The structure maintains $G\subseteq\text{clos}(\text{decompression}(C))$ at all times. MSO₁-testing correctness follows from the equivalence lemma: the satisfaction of $\varphi$ only depends on the finite cabinet type at the root for sufficiently large $S$.

3. Methodological Innovations and Broader Implications

In the aerial measurement context, TreeDGS advances direct object-level metric estimation from RGB imagery by combining explicit 3DGS fields with advanced surface sampling and fit reliability scoring. The integration of depth-aware cumulative opacity sampling distinguishes it from geometry-only point-cloud methods, enabling robust estimation despite viewpoint sparsity and low per-image pixel coverage of trunks (Shaheen et al., 19 Jan 2026). The application of opacity-weighted RANSAC for circle fitting further enhances robustness to outliers and under-observed trunk surfaces.

In the graph-theoretic setting, TreeDGS demonstrates that full dynamization of logical-property testing and structure maintenance is feasible with complexities depending only on tree-depth and formula size, rather than input size. The use of compressed multicabinet representations, driven by MSO formula–characterized structure types, enables theoretical efficiency unattainable with explicit enumeration of all subtrees (Dvorak et al., 2013).

A plausible implication is that both instantiations of "TreeDGS" showcase how careful use of compression—whether of tree-decompositional classes or geometric Gaussians—can facilitate tractable computation and high-fidelity estimation in otherwise intractable or weakly constrained scenarios.

4. Comparative Evaluation

The following table summarizes reported experimental results of the 3DGS-based TreeDGS pipeline versus a LiDAR baseline for aerial DBH measurement (Shaheen et al., 19 Jan 2026):

Method	RMSE (cm)	MAE (cm)	ME (cm)	Success/Total
TreeDGS + opacity-weighted fitting	4.79	3.70	–0.38	189/190
LiDAR + cylinder fitting baseline	7.91	5.04	1.64	187/190

This demonstrates the efficacy of the dense splat-based representation for remote trunk measurement, achieving a 39% reduction in RMSE relative to a state-of-the-art UAV LiDAR system at comparable field-matched sample sizes.

5. Summary and Reproducibility

Both TreeDGS frameworks are fully specified for independent reproduction:

• For aerial DBH estimation: SfM–MVS yields camera poses and a dense point cloud, which initializes a Gaussian field fit via RaDe-GS, followed by dense and reliability-scored sampling, trunk segmentation, and opacity-weighted circle fitting at breast height, finalizing with taper regression to extract DBH.
• For dynamic graph decompositions: A compressed tree-depth decomposition is built with root cabinet MSO₁-flag, supporting vertex/edge updates with per-cabinet recompression and efficient root selection, maintaining $O(1)$ property queries.

TreeDGS thus represents a salient intersection of geometric computer vision and dynamic graph algorithms, each employing compositional compression to achieve state-of-the-art measurement or update efficiency (Shaheen et al., 19 Jan 2026, Dvorak et al., 2013).