3D Hierarchical Proxy Node Representation

• 3D Hierarchical Proxy Node Representation is a method that uses a sparse tree of proxy nodes to capture local geometry, appearance, and motion instead of dense representations.
• It supports efficient neural interpolation and lightweight decoding by propagating coarse-to-fine descriptors for both reconstruction and direct manipulation.
• Experimental evaluations show high-fidelity performance with metrics like PSNR 37.12 and SSIM 0.9858, alongside substantially faster reconstruction times compared to traditional methods.

3D hierarchical proxy node representation denotes a class of 3D encodings in which geometry, appearance, structure, or motion is modeled through a sparse hierarchy of proxy entities rather than through a dense mesh, voxel grid, point cloud, or a single continuous implicit field. In its most explicit formulation, HPR3D represents an object by a tree-structured set of proxy nodes distributed on its surface and interior, where each node stores local shape and texture information and supports efficient neural interpolation, lightweight decoding, and direct manipulation (Wang et al., 16 Jul 2025). Related work applies closely related proxy hierarchies to surface-derived structural trees for navigation-aware abstraction (Andrade et al., 4 Sep 2025), hierarchical motion trees for deformable Gaussian reconstruction (Liang et al., 8 Apr 2025), amino-acid-centered protein graphs with progressively richer geometric proxies (Wang et al., 2022), temporally aligned proxy points for 3D object detection (Chen et al., 2022), and hierarchy-aligned aggregate nodes in three-dimensional OLAP cubes (Brisaboa et al., 2016).

1. Conceptual scope and recurring design pattern

Across the literature, a proxy node is a sparse representative of a local region, local motion, or local aggregate, while hierarchy supplies coarse-to-fine organization. The representation is therefore neither purely explicit nor purely implicit. In HPR3D, the hierarchy is a small tree of proxy nodes over surface and interior; in surface structural extraction, it is a rooted tree over a surface-derived graph; in HiMoR, it is a motion tree in which nodes store relative $\mathbb{SE}(3)$ trajectories; in ProNet, each amino acid is a proxy for local 3D structure and is enriched by backbone and all-atom geometric descriptors; in MPPNet, fixed proxy points inside proposal boxes create temporally aligned object-centric anchors; and in CMHD, internal nodes act as proxy aggregates over hierarchy-defined subcubes rather than geometric regions (Wang et al., 16 Jul 2025, Andrade et al., 4 Sep 2025, Liang et al., 8 Apr 2025, Wang et al., 2022, Chen et al., 2022, Brisaboa et al., 2016).

Work	Proxy carrier	Primary role
HPR3D	Tree-structured proxy nodes on surface and interior	Reconstruction and controllable editing
Structural Tree Extraction	Rooted tree of proxy nodes on a surface-derived graph	Navigation-aware structural analysis
HiMoR	Hierarchical motion tree	Deformation for dynamic Gaussian reconstruction
ProNet	Amino-acid nodes with hierarchical geometric proxies	Protein representation learning
MPPNet	Proxy points inside proposal boxes	Temporal correspondence and feature transport
CMHD	Hierarchy-aligned aggregate nodes in a 3D cube	OLAP summarization

Taken together, these formulations suggest that the phrase names a methodological pattern rather than a single canonical data structure. The shared premise is that a hierarchy of sparse representatives can preserve dominant structure while reducing the burden imposed by dense explicit geometry or by globally entangled latent fields.

2. Canonical node-based formulation in HPR3D

HPR3D formalizes the representation as

$$\mathbf{S} \sim \big((\mathbf{C}^{(1)}, \mathbf{C}^{(2)}, \dots, \mathbf{C}^{(L)}), \theta\big),$$

where each level is a set of proxy points,

$$\mathbf{C}^{(l)} = \{\mathbf{c}^{(l)}_1, \mathbf{c}^{(l)}_2, \dots, \mathbf{c}^{(l)}_{n_l}\}, \quad \mathbf{c}^{(l)}_i = (\mathbf{p}^{(l)}_i, \mathbf{n}^{(l)}_i, \mathbf{f}^{(l)}_i).$$

Here $\mathbf{p}^{(l)}_i$ is node position, $\mathbf{n}^{(l)}_i$ is node normal, and $\mathbf{f}^{(l)}_i$ is a texture or appearance feature. The representation is constructed adaptively from an initial mesh reconstruction: the input can be mesh, point cloud, implicit representation, or multi-view images; the object is converted to a mesh $\mathcal{M} = (\mathbf{V}, \mathbf{F})$; the bottom level is initialized from the mesh vertices $\mathbf{V}$; and higher levels are built by octree partitioning of normalized 3D space together with error-guided clustering (Wang et al., 16 Jul 2025).

Within each non-empty cell at level $l$, HPR3D fits a proxy node to the points in that cell using a local planar least-squares criterion. The fitting error is the curvature-based clustering criterion (CBCC): low error implies that one proxy can summarize the region, while high error implies that the region is too complex and should retain more points at the next level. This yields a tree-structured proxy hierarchy in which higher levels are coarser and cover more points, while lower levels are denser and preserve local detail. Flat or simple regions therefore receive few nodes, whereas curved or detailed regions receive more (Wang et al., 16 Jul 2025).

The significance of this construction lies in its explicit combination of local geometric anchors and learned local descriptors. Coarse nodes control large regions and finer nodes control smaller details, so the hierarchy is not merely a compression device; it is also a control structure. The paper further states that nodes align with local semantics, which underpins its claim of direct drag-and-edit manipulation and scalable quality-complexity control (Wang et al., 16 Jul 2025).

3. Local decoding, reconstruction, and controllable editing

HPR3D decodes appearance by propagating features from coarse to fine levels, fusing local and inherited information, and sending the resulting representation to a small MLP decoder $\phi_\theta$. The fused feature at a lower level depends on the node’s own feature, its parent’s feature, and their relative offset through positional encoding. Inference is therefore local and hierarchical rather than globally field-based: the target described in the paper is mainly mesh-vertex or surface-point reconstruction and rendering, not a single global signed-distance or occupancy function (Wang et al., 16 Jul 2025).

For editing, a displacement $$\mathbf{S} \sim \big((\mathbf{C}^{(1)}, \mathbf{C}^{(2)}, \dots, \mathbf{C}^{(L)}), \theta\big),$$0 applied to a node at level $$\mathbf{S} \sim \big((\mathbf{C}^{(1)}, \mathbf{C}^{(2)}, \dots, \mathbf{C}^{(L)}), \theta\big),$$1 propagates to bottom-level points through a distance-weighted influence,

$$\mathbf{S} \sim \big((\mathbf{C}^{(1)}, \mathbf{C}^{(2)}, \dots, \mathbf{C}^{(L)}), \theta\big),$$2

with weights defined by an exponential decay in the Euclidean distance between a bottom-level point and the edited proxy. After this proxy-guided displacement, the method applies Laplacian editing to produce the final smooth deformation. Coarse edits therefore arise from high-level nodes, while fine edits arise from lower-level nodes. Texture editing follows the same hierarchy: higher-level feature edits affect larger areas, lower-level feature edits affect smaller regions, and texture transfer is implemented by aligning proxy regions, interpolating features between corresponding nodes, and decoding the transferred features (Wang et al., 16 Jul 2025).

The reported experiments evaluate the method on Objaverse with 1,000 textured objects. In the table reproduced in the paper, the proposed method achieves PSNR $$\mathbf{S} \sim \big((\mathbf{C}^{(1)}, \mathbf{C}^{(2)}, \dots, \mathbf{C}^{(L)}), \theta\big),$$3 and SSIM $$\mathbf{S} \sim \big((\mathbf{C}^{(1)}, \mathbf{C}^{(2)}, \dots, \mathbf{C}^{(L)}), \theta\big),$$4, while MASH is slightly better on Chamfer Distance. The reported reconstruction time is $$\mathbf{S} \sim \big((\mathbf{C}^{(1)}, \mathbf{C}^{(2)}, \dots, \mathbf{C}^{(L)}), \theta\big),$$5s for joint geometry+texture reconstruction, compared with $$\mathbf{S} \sim \big((\mathbf{C}^{(1)}, \mathbf{C}^{(2)}, \dots, \mathbf{C}^{(L)}), \theta\big),$$6h for MASH geometry optimization in the reported setup. The geometry-only parameter count is around $$\mathbf{S} \sim \big((\mathbf{C}^{(1)}, \mathbf{C}^{(2)}, \dots, \mathbf{C}^{(L)}), \theta\big),$$7, and the joint geometry+texture representation uses $$\mathbf{S} \sim \big((\mathbf{C}^{(1)}, \mathbf{C}^{(2)}, \dots, \mathbf{C}^{(L)}), \theta\big),$$8, compared with $$\mathbf{S} \sim \big((\mathbf{C}^{(1)}, \mathbf{C}^{(2)}, \dots, \mathbf{C}^{(L)}), \theta\big),$$9 and $$\mathbf{C}^{(l)} = \{\mathbf{c}^{(l)}_1, \mathbf{c}^{(l)}_2, \dots, \mathbf{c}^{(l)}_{n_l}\}, \quad \mathbf{c}^{(l)}_i = (\mathbf{p}^{(l)}_i, \mathbf{n}^{(l)}_i, \mathbf{f}^{(l)}_i).$$0 for FlexiCubes in the reported setting. These results support the paper’s description of the representation as compact, high-fidelity, and directly editable (Wang et al., 16 Jul 2025).

4. Surface-derived hierarchical proxy trees

A distinct but closely related formulation appears in structural tree extraction from 3D surfaces. Here the core contribution is a surface-driven hierarchical abstraction pipeline that transforms an unorganized polygonal mesh into a rooted tree of proxy nodes suitable for navigation-aware structural analysis. The full procedure is formalized as Algorithm 1 and consists of five stages: building an initial graph from the input geometry; identifying terminal nodes according to the target application; computing a Steiner tree connecting those terminals in a cost-efficient way; converting the unrooted Steiner tree into one or more rooted trees; and simplifying the rooted trees using line-of-sight and geometric consistency constraints (Andrade et al., 4 Sep 2025).

The initial graph is defined as a superset of all potential solution graphs, and its construction depends on the use case. For navigation or map analysis, the graph is built directly from the navigation mesh: each node corresponds to a convex planar polygon, edges are undirected adjacency links between neighboring polygons, and edge weights are Euclidean distances between associated node positions. This preserves walkable connectivity, topological adjacency of traversable regions, and a geometry-aware notion of distance along the surface, while discarding non-walkable structures, overhangs, vertical details, and small surface features disconnected from the ground. When no navmesh is available, or finer granularity is required, the graph can instead be built by sampling the surface and connecting sampled points through a neighborhood relation whose weights may encode Euclidean distance, material properties, geometric features, or visibility and line of sight (Andrade et al., 4 Sep 2025).

Terminal selection is application-specific. For modular map-piece extraction, terminals are derived from entry and exit points on the geometry boundary by extracting boundary edges, sampling boundary points at regular intervals, classifying those points by local raycasting, tracing boundary segments, and matching the segments back to the original geometry. For map analysis, terminal selection can instead be driven by graph-theoretic metrics including degree centrality, betweenness centrality, eigenvector centrality, and Katz centrality. In the reported experiments, entry and exit points serve as terminals for map-piece extraction, while leaf nodes serve as terminals for level analysis because dead ends help cover the level (Andrade et al., 4 Sep 2025).

The Steiner stage uses the Kou et al. approximation: build a complete graph over the terminals, compute an MST, replace each MST edge by the corresponding shortest path in the original graph, insert all intermediate nodes and edges into a new graph, compute a second MST, and remove isolated zero-degree nodes. Kruskal’s algorithm is used for MST computation. The rooted trees are then simplified by collapsing degree-2 chains $$\mathbf{C}^{(l)} = \{\mathbf{c}^{(l)}_1, \mathbf{c}^{(l)}_2, \dots, \mathbf{c}^{(l)}_{n_l}\}, \quad \mathbf{c}^{(l)}_i = (\mathbf{p}^{(l)}_i, \mathbf{n}^{(l)}_i, \mathbf{f}^{(l)}_i).$$1 subject to a predicate that checks navmesh validity for leaf nodes, tolerance on the angle between surface normals, and line-of-sight constraints. This pruning removes unnecessary degree-2 nodes, reduces path redundancy, and preserves essential connectivity and surface coherence (Andrade et al., 4 Sep 2025).

The resulting object is a hierarchical proxy graph for 3D surface structure rather than a physical skeleton inside the object. Nodes are proxy landmarks on the surface, edges summarize navigable or structurally meaningful connections, and tree depth encodes structure relative to a chosen root. The method is validated on tile-based elements from the “Procedural Generation of 3D Maps With Snappable Meshes” dataset and on full game levels including “3D Scifi Kit Starter Kit” and “Blue Dungeon.” The paper also states a central limitation: because the result is a tree, loops and cyclic routes are suppressed, and cyclic areas can be lost or underrepresented (Andrade et al., 4 Sep 2025).

5. Task-specific extensions across motion, detection, proteins, and hierarchical domains

HiMoR applies hierarchical proxy organization to motion rather than static geometry. It models motion as a tree graph in which each node stores a motion sequence in $$\mathbf{C}^{(l)} = \{\mathbf{c}^{(l)}_1, \mathbf{c}^{(l)}_2, \dots, \mathbf{c}^{(l)}_{n_l}\}, \quad \mathbf{c}^{(l)}_i = (\mathbf{p}^{(l)}_i, \mathbf{n}^{(l)}_i, \mathbf{f}^{(l)}_i).$$2 relative to its parent node, with the root fixed at the world origin. Non-leaf nodes provide a rough motion foundation, leaf nodes provide the finest motion estimates, and motions under a common parent are represented by a few shared motion bases. In the experiments, HiMoR uses a two-layer hierarchy excluding the root: the first layer starts with 50 nodes, and after densification each first-layer node spawns 10 child nodes; the first layer shares 10 motion bases and the second layer shares 5 motion bases. Gaussian deformation is interpolated from nearby leaf nodes, dual quaternions are used for interpolation quality, and rigidity regularization is stronger for shallower nodes and weaker for deeper nodes, enforcing smooth coarse motion while leaving room for finer local detail (Liang et al., 8 Apr 2025).

MPPNet uses a related but not identical proxy-point formulation for 3D temporal object detection. For each proposal trajectory, it samples $$\mathbf{C}^{(l)} = \{\mathbf{c}^{(l)}_1, \mathbf{c}^{(l)}_2, \dots, \mathbf{c}^{(l)}_{n_l}\}, \quad \mathbf{c}^{(l)}_i = (\mathbf{p}^{(l)}_i, \mathbf{n}^{(l)}_i, \mathbf{f}^{(l)}_i).$$3 proxy points uniformly inside each proposal box, with $$\mathbf{C}^{(l)} = \{\mathbf{c}^{(l)}_1, \mathbf{c}^{(l)}_2, \dots, \mathbf{c}^{(l)}_{n_l}\}, \quad \mathbf{c}^{(l)}_i = (\mathbf{p}^{(l)}_i, \mathbf{n}^{(l)}_i, \mathbf{f}^{(l)}_i).$$4 used as a good trade-off. Because the proxy points are sampled in the same relative way in every frame, proxy point $$\mathbf{C}^{(l)} = \{\mathbf{c}^{(l)}_1, \mathbf{c}^{(l)}_2, \dots, \mathbf{c}^{(l)}_{n_l}\}, \quad \mathbf{c}^{(l)}_i = (\mathbf{p}^{(l)}_i, \mathbf{n}^{(l)}_i, \mathbf{f}^{(l)}_i).$$5 in frame $$\mathbf{C}^{(l)} = \{\mathbf{c}^{(l)}_1, \mathbf{c}^{(l)}_2, \dots, \mathbf{c}^{(l)}_{n_l}\}, \quad \mathbf{c}^{(l)}_i = (\mathbf{p}^{(l)}_i, \mathbf{n}^{(l)}_i, \mathbf{f}^{(l)}_i).$$6 corresponds to the same semantic part of the object as proxy point $$\mathbf{C}^{(l)} = \{\mathbf{c}^{(l)}_1, \mathbf{c}^{(l)}_2, \dots, \mathbf{c}^{(l)}_{n_l}\}, \quad \mathbf{c}^{(l)}_i = (\mathbf{p}^{(l)}_i, \mathbf{n}^{(l)}_i, \mathbf{f}^{(l)}_i).$$7 in any other frame. MPPNet organizes temporal encoding in three hierarchies: per-frame feature encoding, short-clip or group-level fusion, and whole-sequence aggregation. The proxy points are described explicitly as consistent object representations and as couriers that facilitate feature interaction between frames, making long-sequence temporal fusion more tractable than direct all-frame raw-point concatenation (Chen et al., 2022).

ProNet keeps graph nodes at the amino-acid level but enriches each node with hierarchical geometric proxies so that the same protein can be represented at amino-acid, backbone, and all-atom levels. Each amino acid is therefore a proxy for a local 3D structure, and progressively richer descriptors are attached to the same graph rather than through conventional pooling. The paper proves that if the base 3D graph representation is complete, then the hierarchical geometric representations are also complete at all levels. It also reports that different downstream tasks prefer different levels: backbone representations are usually strongest for fold classification and reaction classification, while all-atom representations often perform best for protein-ligand binding affinity and protein-protein interaction (Wang et al., 2022).

CMHD extends the proxy idea to three-dimensional hierarchical domains in OLAP rather than geometric reconstruction. In its 3D specialization, each conceptual node corresponds to a Cartesian product of one node from each of three dimension hierarchies, and the node stores the aggregate value for the corresponding subcube. Internal nodes therefore serve as proxy aggregates for cross-dimensional hierarchy blocks. The structure stores the hierarchy trees with LOUDS and the conceptual tree with $$\mathbf{C}^{(l)} = \{\mathbf{c}^{(l)}_1, \mathbf{c}^{(l)}_2, \dots, \mathbf{c}^{(l)}_{n_l}\}, \quad \mathbf{c}^{(l)}_i = (\mathbf{p}^{(l)}_i, \mathbf{n}^{(l)}_i, \mathbf{f}^{(l)}_i).$$8, $$\mathbf{C}^{(l)} = \{\mathbf{c}^{(l)}_1, \mathbf{c}^{(l)}_2, \dots, \mathbf{c}^{(l)}_{n_l}\}, \quad \mathbf{c}^{(l)}_i = (\mathbf{p}^{(l)}_i, \mathbf{n}^{(l)}_i, \mathbf{f}^{(l)}_i).$$9, and a value array $\mathbf{p}^{(l)}_i$0. Its main distinction from a generic $\mathbf{p}^{(l)}_i$1-treap is that the partition follows hierarchy boundaries rather than regular geometric subdivision, so hierarchical summary queries can be resolved by aggregating fewer tree nodes (Brisaboa et al., 2016).

6. Strengths, assumptions, and recurrent limitations

Several strengths recur across these formulations. HPR3D explicitly seeks the structural clarity and editability of explicit control points together with the compactness and reconstruction fidelity of learned local features. Structural tree extraction is surface-first and navigation-aware, and can adapt terminal selection to entry and exit extraction or graph metrics. HiMoR decomposes motion into coarser motions and finer details, using shared motion bases and level-dependent rigidity to improve temporal smoothness. ProNet preserves hierarchical structural information while keeping the graph size at the amino-acid level rather than the atom level. MPPNet makes long-sequence temporal modeling computationally manageable by replacing irregular, frame-varying point sets with a stable, aligned set of object-centric proxies. CMHD answers hierarchy-aligned summary queries by reusing precomputed aggregates rather than traversing a generic quadtree-like partition (Wang et al., 16 Jul 2025, Andrade et al., 4 Sep 2025, Liang et al., 8 Apr 2025, Wang et al., 2022, Chen et al., 2022, Brisaboa et al., 2016).

The assumptions are equally recurrent. Proxy methods assume that relevant structure can be localized to sparse representatives and that hierarchy is a meaningful inductive bias. In HPR3D, this means that geometry and appearance can be captured by local node features decoded by a small MLP; in structural extraction, that shortest-path connectivity over a surface graph is an adequate proxy for structural relation and that a tree is acceptable when loops are not essential; in HiMoR, that motion is low-rank and compositional; in ProNet, that amino-acid nodes can act as complete proxies when enriched by the appropriate geometric descriptors; in MPPNet, that fixed relative box coordinates provide stable cross-time correspondence; and in CMHD, that query workloads follow hierarchy nodes rather than arbitrary multidimensional ranges (Wang et al., 16 Jul 2025, Andrade et al., 4 Sep 2025, Liang et al., 8 Apr 2025, Wang et al., 2022, Chen et al., 2022, Brisaboa et al., 2016).

The limitations are correspondingly domain specific. HPR3D depends on an initial mesh reconstruction, uses an MLP-based texture decoder that may remain challenged by highly non-local or view-dependent appearance, still relies on Laplacian post-processing for geometry editing, and is not presented as a universal all-space volumetric query representation. Surface structural extraction excludes loops by construction, depends heavily on the initial graph, inherits navmesh bias toward walkable surfaces, is sensitive to mesh artifacts and normal misalignment, and may produce counterintuitive Steiner minimality relative to human expectations of central hubs. ProNet shows that different tasks may require different hierarchy levels rather than a single universally best one. CMHD assumes hierarchies of the same height, requires extra space for LOUDS and $\mathbf{p}^{(l)}_i$2, and is not intended for arbitrary range-query workloads. A plausible implication is that proxy hierarchies are most effective when the target task already admits a semantically meaningful decomposition into coarse structure and local refinement (Wang et al., 16 Jul 2025, Andrade et al., 4 Sep 2025, Wang et al., 2022, Brisaboa et al., 2016).