Point Cloud Serialization

• Point Cloud Serialization is a method that converts unordered 3D points into structured 1D sequences while preserving spatial proximity.
• Key techniques such as Morton (Z-order), Hilbert, and adaptive grid-based orderings balance computational cost with improved locality preservation.
• Effective serialization enhances neural sequence models, leading to measurable gains in tasks like classification, segmentation, and compression.

Point cloud serialization refers to the process of mapping an unordered, irregular set of 3D points into a one-dimensional (1D) sequence or set of sequences with the goal of preserving spatial locality and semantic structure—thereby enabling structured neural sequence processing of geometric data. Serialization is foundational for employing sequence models such as State Space Models (SSMs), Transformers, or neural compression architectures, and serves as a bridge between set-based 3D representations and efficient, high-capacity sequential modeling.

1. Motivation and Conceptual Foundations

Point clouds $\mathcal{P} = \{ x_i \in \mathbb{R}^3 \}_{i=1}^N$ are intrinsically unordered, lacking native sequential or grid structure. However, state-of-the-art sequence models—including SSMs (e.g., Mamba) and Transformers—expect structured 1D token sequences. Direct application of generic or arbitrary orderings (e.g., by $x$-coordinate, random permutation) disrupts spatial adjacency and undermines the ability of linear convolutional kernels or causal models to capture local geometric relationships. Effective serialization must therefore establish a mapping $f:\mathbb{R}^3\to\mathbb{R}$, or $f: \mathcal{P} \to \{s_i\}_{i=1}^N$, such that spatial neighbors remain sequence neighbors ($||x_i - x_j||_2$ small $\implies$ $|s_i - s_j|$ small) (Liu et al., 16 Jun 2025, Zhang et al., 2024).

This locality-preserving property underpins both the computational efficiency and the geometric fidelity of modern point cloud learning frameworks.

2. Serialization via Space-Filling Curves and Grid-Based Orderings

A core class of serialization strategies leverages space-filling curves (SFCs), which define explicit traversals of a quantized 3D grid, mapping points to indices that reflect spatial proximity. Common SFCs include:

• Morton (Z-order) Curve: Bits of $x$, $y$, $z$ integer coordinates are interleaved to produce a single code $x$0, where $x$1 are bits of the quantized coordinates. Sorting points by $x$2 yields the Z-order sequence, widely used for efficient neighbor lookups and as input to linear-complexity sequence models (Liu et al., 16 Jun 2025).
• Hilbert and Trans-Hilbert Curves: These recursively traverse 3D grids with maximal neighborhood preservation, defined through Gray-code mapped subcubes and orientation state updates. Sorting points by 3D Hilbert indices provides stronger locality than Z-order, which is reflected in higher recall/segmentation accuracy in numerous empirical ablations (Qu et al., 26 Jul 2025, Lin et al., 23 Jul 2025).
• Consistent Traverse Serialization (CTS): A "snake" scan traverses quantized voxels in a back-and-forth pattern along sequential axes; by permuting axes, multiple equivalent traversals are defined (e.g., XYZ, YXZ, etc.), each providing a unique neighborhood mapping. These traversals can be alternated across model layers to comprehensively expose the network to all spatial adjacencies (Zhang et al., 2024).

Table: Properties of Key SFC Serializations

Scheme	Locality Preservation	Computational Cost
Single axis sort	Poor	$x$3
Morton (Z-order)	Moderate	$x$4
Hilbert curve	Very good	$x$5
CTS (snake+perm)	Good (multi-view)	$x$6

3. Task-Aware and Adaptive Serialization Strategies

The optimal serialization method is task-dependent:

• Classification: Global structure is best captured by a single, locality-maximizing SFC ordering, e.g., Hilbert or Z-order. Empirical results indicate that Hilbert or Trans-Hilbert yields PB-T50-RS $x$7 on ScanObjectNN versus $x$8 for random (Lin et al., 23 Jul 2025).
• Segmentation: Local continuity is crucial. Axis-wise sorting (sorting and concatenating sequences along $x$9, $f:\mathbb{R}^3\to\mathbb{R}$0, $f:\mathbb{R}^3\to\mathbb{R}$1) delivers superior segmentation mIoU ($f:\mathbb{R}^3\to\mathbb{R}$2 on ShapeNetPart vs. $f:\mathbb{R}^3\to\mathbb{R}$3 for Hilbert), as each axis-sorted subsequence groups local neighborhoods relevant for fine-grained part segmentation (Lin et al., 23 Jul 2025).
• Structure-Adaptive Serialization: DM3D introduces a learnable, sample-adaptive approach, Offset-Guided Gaussian Sequencing, that predicts per-point spatial and sequence offsets to reorder and resample points dynamically based on structure, rather than following a fixed SFC. The Gaussian-based KNN resampling and differentiable reordering steps allow end-to-end optimization of serialization. This adaptive process yields up to $f:\mathbb{R}^3\to\mathbb{R}$4 accuracy improvement in classification versus static Hilbert baselines (Liu et al., 3 Dec 2025).

4. Architectural Integration and Complexity

Serialized sequences are employed as input streams to sequence models:

• Mamba (SSM) Blocks: After serialization, the sequence is passed to SSM-based modules, either in forward, backward, or bidirectional scan modes. Complexity is linear in sequence length (e.g., $f:\mathbb{R}^3\to\mathbb{R}$5 for SSM/Mamba, compared to $f:\mathbb{R}^3\to\mathbb{R}$6 for self-attention), enabling processing of large-scale data with tractable memory and compute profiles (Liu et al., 16 Jun 2025, Zhang et al., 2024).
• Axis-wise and Multi-view Fusion: Methods such as CloudMamba process each axis-sorted sequence independently via Mamba, then merge feature outputs (sequence merging), yielding stereoscopic geometry insight. The "chainedMamba" variant fuses forward and backward passes to better capture high-level geometric context (Qu et al., 11 Nov 2025).
• Causal and Multi-head Variants: Shuffle serialization (HydraMamba) applies different SFC variants at each layer, injecting multiple geometric perspectives and preventing overfitting to a single scan direction. This strategy enhances generalization and enables causal SSMs to exploit richer adjacency patterns (Qu et al., 26 Jul 2025).
• Positional Embedding: Coordinate-based or window-position embeddings are often added to serialized features prior to sequence modeling, further preserving spatial proximity and resolving ambiguities introduced by quantization or serialization (Zhang et al., 2024).

5. Serialization in Compression, Registration, and Reconstruction Pipelines

Efficient serialization is not restricted to learning paradigms:

• Compression: Multiscale sparse convolutional autoencoders compress point clouds by downsampling and serializing geometry at bottleneck layers as octree codes or via context-encoded feature vectors. Geometry is typically serialized using octree (Morton-order), and features are compressed with learned probabilistic models (Wang et al., 2020).
• Surface Reconstruction: Methods like NoKSR serialize points via Hilbert codes at multiple grid scales to enable fast approximate neighbor lookups for SDF regression. Multi-scale serialization mitigates recall loss, ensuring that missed neighbors at fine levels are captured at coarser ones (Li et al., 18 Feb 2025).
• Registration: MT-PCR applies Z-order serialization, quantizing points and sorting by Morton code, to precondition features before Mamba encoding and Transformer-based refinement. The removal of explicit positional indicators further improves registration accuracy, with Z-order serialization alone providing a $f:\mathbb{R}^3\to\mathbb{R}$7 absolute recall boost (Liu et al., 16 Jun 2025).

6. Ray-Aligned and Sensor-Centric Serialization

LiDAR-based 3D detection and compression introduces sensor-centric serialization strategies:

• Ray-Aligned Serialization: RayMamba groups voxels into azimuthal sectors, sorting within sectors by vertical position and angular residue. This sector-wise, directionally coherent serialization encodes occlusion and scan continuity, yielding up to $f:\mathbb{R}^3\to\mathbb{R}$8 mAP and $f:\mathbb{R}^3\to\mathbb{R}$9 NDS improvement on nuScenes at $f: \mathcal{P} \to \{s_i\}_{i=1}^N$0–$f: \mathcal{P} \to \{s_i\}_{i=1}^N$1 range (Lu et al., 3 Apr 2026).
• Scan-Order Serialization for LiDAR: In reflectance compression (SerLiC), points are discretized into laser beam and azimuth bins (as in sensor acquisition), then ordered within each beam. This scan-order serialization is tightly aligned with device physics, facilitating efficient neural compression and reflecting intrinsic temporal/autoregressive dependency at minimal model size (Zhu et al., 14 May 2025).

7. Quantitative Impact and Ablation Analysis

Empirical ablation studies consistently validate the importance of serialization:

• Conversion from random or single-axis serialization to SFC-based ordering typically yields +2–7% improvement in critical classification/segmentation metrics across domains (Liu et al., 16 Jun 2025, Lin et al., 23 Jul 2025, Qu et al., 11 Nov 2025).
• Adaptive, deformable, or multi-view serialization achieves further moderate gains (+1–3%) in challenging settings where geometric diversity or task structure demands more flexible adjacency patterns (Liu et al., 3 Dec 2025, Qu et al., 26 Jul 2025).
• Complexity remains linear in $f: \mathcal{P} \to \{s_i\}_{i=1}^N$2 (sequence length) for all SSM/Mamba and multi-SFC models, with memory and runtime dominated by batch, channel, and model size, not by serialization order (Zhang et al., 2024, Zhang et al., 2024).

In summary, point cloud serialization is a fundamental enabler for high-capacity, locality-preserving point cloud learning and compression. The design and tuning of serialization strategies—ranging from fixed space-filling curves to adaptive, structure-aware orderings—directly impact both computational efficiency and the attainable upper bound of geometric reasoning in modern 3D neural architectures.