LiteGE: Lightweight Geodesic Estimation

• LiteGE is a lightweight neural framework that uses PCA-based shape descriptors derived from UDF samples for rapid geodesic distance estimation.
• It achieves dramatic memory and speed improvements—up to 100–1000× faster inference—by replacing large 3D backbones with a compact representation.
• The unified design supports both geodesic regression and non-isometric shape correspondence, enabling real-time applications in AR/VR, robotics, and on-device 3D vision.

LiteGE is a lightweight neural framework for efficient geodesic distance estimation and non-isometric shape correspondence on 3D surfaces, designed to operate on both meshes and point clouds. It replaces large 3D backbone architectures with compact, category-aware shape descriptors derived from unsigned distance field (UDF) samples and their principal component analysis (PCA) projections. This architecture delivers orders-of-magnitude improvements in memory usage and inference speed while maintaining high accuracy, even for sparse or noisy 3D inputs (Adikusuma et al., 19 Dec 2025).

1. Motivation and Context

Classical mesh-based geodesic solvers compute geodesic distances between pairs of surface points with exact or approximate algorithms, typically incurring minutes to hours of runtime on large meshes. Learning-based neural approaches (e.g., NeuroGF, GeGNN) significantly reduce per-query latency to the millisecond scale but require multi-gigabyte 3D networks, which are impractical for real-time or memory-constrained scenarios and exhibit poor performance on sparse point clouds.

Many applications in AR/VR, robotics, and on-device 3D vision demand rapid geodesic queries across many shapes with limited compute and memory. Additionally, there is a tight connection between geodesic distance prediction and finding correspondences between non-isometric shapes—yet this relationship remained under-exploited by previous work. LiteGE directly addresses these gaps by substituting neural 3D backbones with an ultra-compact representation, yielding interactive runtimes and robust generalization across point density and shape types.

2. Category-Aware Compact Shape Descriptor

The shape descriptor pipeline consists of canonicalization, voxelization, informative voxel selection, UDF sampling, and PCA projection:

a. Shape Canonicalization

• The input shape is translated to place its centroid at the origin.
• The surface is scaled to unit area for meshes or to a fixed bounding-box area for point clouds.
• A consistent orientation is enforced via PCA axes (for regression tasks) or by a learned T-Net (for matching scenarios).

b. Voxelization and Informative Voxels

• Canonicalized shape is voxelized in a $128^3$ grid, labeling voxels as inside ($1$) or outside ($0$) the shape.
• Voxel occupancy variance is computed over the training set. Voxels with variance exceeding a threshold $\epsilon$—typically those near the surface—are retained, reducing the representation to $\sim20{,}000$–$30{,}000$ "informative" voxels.

c. UDF Sampling and PCA Embedding

• For each selected voxel center $p$, the UDF $x_{\mathrm{UDF}}(p)$ is the unsigned distance to the nearest point on the shape surface.
• All UDF values are concatenated to yield a high-dimensional vector $x_{\mathrm{UDF}}$ ($M \approx 20{,}000$).
• PCA is performed on the $x_{\mathrm{UDF}}$ vectors across the training set. The top $D$ principal components are retained; $D=50$ components typically explain over 95% variance.
• The final shape descriptor is $z_{\mathrm{UDF-PCA}} \in \mathbb{R}^D$ for each shape, with $$z_i = \langle x_{\mathrm{UDF}} - \mu_{\mathrm{UDF}}, p_i\rangle$$ where $\mu_{\mathrm{UDF}}$ is the mean UDF vector and $p_i$ are the principal components.

3. Algorithmic Pipeline and Inference

The LiteGE workflow is divided into category-level preprocessing and per-shape inference:

A. Preprocessing (per Shape Category)

1. Collect and canonicalize training shapes; voxelize each.
2. Identify informative voxels and compute $x_{\mathrm{UDF}}$ vectors.
3. Fit PCA to obtain $\mu_{\mathrm{UDF}}$ and principal components.

B. Inference (per Shape)

1. Canonicalize the input shape.
2. For each informative voxel center, compute UDF.
3. Project UDF vector to PCA space to obtain $z_{\mathrm{UDF-PCA}}$.
4. Feed $z_{\mathrm{UDF-PCA}}$ through a small MLP (three layers, $200$ units each) to obtain a global shape embedding.
5. Embed query points $s$, $d \in \mathbb{R}^3$ via a two-layer Coord-MLP ($200$ units each), concatenate with global shape descriptor, and pass through a shared three-layer point MLP (also $200$ units each).
6. The difference of the two point embeddings is input to a final two-layer MLP (each $400$ units) to predict the geodesic $G(s,d; z_{\mathrm{UDF-PCA}})$.

C. Support for Sparse Inputs

Because the UDF descriptor is independent of sampling density, LiteGE operates with inputs as sparse as $300$ points, a regime in which prior neural backbones (e.g., NeuroGF) fail.

4. Computational Complexity and Empirical Efficiency

The LiteGE architecture offers large improvements in runtime and memory usage:

System	Parameters / Memory	Inference Time (1 query)	Inference Time (64 queries)	Shape Matching Time (5K verts)
NeuroGF	>10 GB, multi-GPU	88 ms	1529 ms	N/A
LiteGE	≈600K / <50 MB	1.9 ms (47× faster)	13.4 ms (114× faster)	119 ms (≈1000× faster)

• PCA projection costs $\mathcal{O}(D \cdot M)$ per shape, with $D \ll M$.
• Nearest-neighbor UDF computation: $\mathcal{O}(M \log N)$ for $N$ surface samples.
• MLP execution is $\mathcal{O}($number of parameters$)$.
• Overall, LiteGE yields up to $300\times$ memory reduction and 100–300$\times$ speedup over neural methods, and $100$–$1000\times$ speedup over mesh-based correspondence solvers (Adikusuma et al., 19 Dec 2025).

5. Geodesic Distance and Non-Isometric Shape Correspondence

LiteGE is trained solely on intra-shape geodesic regression, yet at inference it supports shape-to-shape correspondence via an “average” cross-shape geodesic:

$$G_{XY}(s_X, p_Y) = \tfrac{1}{2}\left[G(s_X, p_Y; z_X) + G(s_X, p_Y; z_Y)\right]$$

Matching a query point $q_X$ to a target shape $Y$ is formulated as: $p^* = \arg \min_{p \in Y} G_{XY}(q_X, p)$

To avoid brute-force $O(|V|)$ evaluation, LiteGE organizes the search with a multi-tier nearest-neighbor cache, starting from a sparse anchor set and recursively refining to all vertices for sub-millisecond per-point matching. This design supports interactive landmark and region correspondence and generalizes to non-isometric shape pairs and incomplete data.

6. Experimental Results

a. Geodesic Regression (normalized; mean distance $=100$):

• 300-pt clouds: LiteGE $L_1 = 3.4$; NeuroGF $=75$ (failure).
• 2000-pt clouds: LiteGE $= 2.3$; NeuroGF $= 13$.
• Generalization: FAUST (700 pts) L₁ = 2.6, Objaverse-XL (2000 pts) L₁ = 3.9.

b. Shape Matching (1000 matches, error$\times100$):

• 4K-vertex clean mesh: SMS AUC 79.4%, LiteGE 79.3%; mean error 2.2 vs 2.5; runtimes 21.3s vs 0.185s.
• 5K remeshed mesh: SMS error 7.44, LiteGE 7.2; 27s vs 0.119s.
• Broken mesh (40% faces removed): SMS error >28 (AUC 33%), LiteGE error 8 (AUC 69.5).
• Point clouds (8K/500 pts): error 7.6/8.1 in ≈200/126 ms.

c. Geodesic Path Tracing:

On clouds $<$1,000 points, LiteGE recovers paths with median L₁ errors 3.7–4.7 (scaled to mean=100) in $<$6 MB memory.

7. Advantages, Limitations, and Applications

Advantages

• Extreme memory and runtime efficiency ($\leq$50 MB, single-digit ms inference).
• Robustness to sparse, noisy, or incomplete input.
• Unification of geodesic and non-isometric matching tasks in a single framework.
• No reliance on large 3D neural backbones, facilitating edge deployment.

Limitations

• Requires category-specific canonicalization for maximum accuracy.
• PCA basis and informative voxel set are category-specific, incurring upfront precomputation cost.
• Very fine surface details may be lost if voxel resolution or PCA dimension is low.

Applications

• Real-time shape editing, sculpting, and texture mapping (fast geodesic computation).
• Landmark and region matching in AR/VR and robotics with sparse sensing.
• Extension to dynamic 4D shapes with temporal consistency (with further development).
• On-device 3D vision tasks (SLAM, segmentation, retrieval) under memory constraints.

LiteGE demonstrates that a compact PCA embedding of UDF samples is sufficient to replace high-capacity neural 3D backbones for both geodesic regression and non-isometric correspondence with two to three orders of magnitude gains in efficiency while preserving competitive accuracy (Adikusuma et al., 19 Dec 2025).