Light Detection And Ranging (LiDAR) point cloud data has become increasingly prevalent in various applications including autonomous vehicles, robotics, augmented reality, and digital preservation [1]. These data, often acquired through LiDAR scanners or structured light systems, typically contain millions to billions of points, presenting significant challenges for storage, transmission, and processing [2–5]. Effective point cloud down-sampling (simplification) algorithms are therefore essential for practical applications. Existing simplification methods can be broadly categorized into several types: geometry-based methods (such as Curvature-aware Sampling, CS), grid-based methods (like Voxel Grid down-sampling, VG), clustering-based approaches, and learning-based techniques [6]. While each category has its strengths, most suffer from either high computational complexity or inadequate feature preservation, particularly when dealing with complex geometric structures [7].

Early point cloud simplification techniques primarily focused on spatial uniformity [8]. Random Sampling (RS) provides the simplest approach but often fails to preserve important geometric features [9]. VG down-sampling offers improved uniformity but tends to smooth sharp features and edges [10]. CS methods address this limitation by prioritizing areas with high geometric variation, though they require expensive normal and curvature estimations [11]. Clustering methods group spatially proximate points and select representative points from each cluster. K-means and DBSCAN have been widely used, but they typically operate in the full three-dimensional space, resulting in high computational costs for large datasets [12]. Some approaches have explored hierarchical clustering to reduce complexity, but these often sacrifice feature preservation at fine scales [13].

Recent advances in deep learning have led to point cloud simplification networks that learn task-specific importance metrics. While promising, these methods require extensive training data and computational resources, limiting their applicability in resource-constrained environments [14–19]. Fuzzy C-Means (FCM) clustering has been applied to point cloud segmentation and classification due to its ability to handle uncertainty in data membership [20]. However, its application to point cloud simplification has been limited, primarily due to the high computational cost of three-dimensional FCM clustering on large datasets [21].

This paper presents a novel down-sampling method that overcomes these limitations via a unique dimensional decomposition approach. By independently applying FCM clustering to each spatial dimension and integrating the resulting membership information, our method attains an optimal equilibrium between computational efficiency and geometric fidelity. The fundamental innovation of our proposed methodology lies in a systematic dimensional decomposition strategy that transforms the inherently complex three-dimensional point cloud simplification challenge into a series of more manageable one-dimensional analytical problems. This paradigm shift allows us to utilize the mathematical elegance and computational efficiency of single-dimensional fuzzy clustering while still capturing the essential three-dimensional geometric characteristics of point cloud data.

At the core of our approach lies the independent application of FCM clustering to each spatial coordinate axis. This process generates distinct membership functions for the X, Y, and Z dimensions, each representing the relationship between points and the distributional characteristics along that specific axis. These membership functions act as mathematical descriptors that quantify the extent to which each point represents the underlying statistical patterns within each dimension. The dimensionality reduction inherent in this approach provides substantial computational benefits, as the complexity of FCM clustering scales more favorably when applied to one-dimensional data than to its three-dimensional counterpart.

The subsequent integration of these multi-dimensional membership values forms the second crucial innovation of our method. Instead of treating the three dimensions as distinct entities, we have developed a sophisticated integration framework that combines the dimensional information into a unified importance metric. This integration process uses a product-based formulation that inherently highlights points that demonstrate high representativeness across all three dimensions simultaneously. Such points usually correspond to locations where the point cloud displays characteristic geometric features, whereas points with low multi-dimensional representativity often signify redundant or noisy data that can be safely removed without sacrificing geometric fidelity.

The adaptive sampling strategy driven by the derived importance scores achieves a principled balance among the competing objectives inherent in point cloud simplification. Points located at geometrically salient regions are assigned higher importance values due to their strong representativeness across multiple dimensions. Such points typically delineate surface boundaries, characterize curvature extrema, or indicate structural intersections. Conversely, points residing in homogeneous areas or arising from measurement noise exhibit weak dimensional coherence and are therefore assigned lower importance scores. This systematic importance evaluation enables the proposed algorithm to selectively retain critical geometric information while rigorously discarding redundant data. As a result, the method delivers an intelligent simplification capability that surpasses conventional uniform or randomly weighted sampling schemes. In essence, sampling is transformed from a passive selection procedure into an active, information-aware decision-making process that explicitly respects the intrinsic geometric structure of point cloud data.

Moreover, the dimensional independence of our analysis provides inherent robustness to anisotropic data distributions commonly encountered in practical applications. Real-world LiDAR acquisitions often exhibit varying point densities along different axes due to scanner characteristics or environmental factors. Our method naturally accommodates such variations by analyzing each dimension according to its own statistical properties, without imposing artificial symmetry constraints that might distort the simplification process. The theoretical underpinning of this approach rests on the observation that while point clouds exist in three-dimensional space, many of their essential geometric characteristics manifest distinctly along individual coordinate axes. Sharp edges, for instance, typically involve rapid changes in two dimensions while maintaining consistency in the third. Planar surfaces exhibit statistical regularity along their normal direction while showing diversity in the tangential dimensions. By capturing these dimensional signatures through independent FCM clustering and subsequently synthesizing them through intelligent fusion, our method achieves a comprehensive understanding of point cloud geometry that informs effective simplification decisions.

Furthermore, the dimensional independence underlying the proposed analysis confers inherent robustness to anisotropic point distributions frequently encountered in real-world scenarios. LiDAR acquisitions often exhibit nonuniform sampling densities across different axes due to sensor configurations or environmental influences. By treating each dimension according to its own statistical characteristics, the proposed approach naturally accommodates such anisotropy without enforcing artificial symmetry assumptions that could compromise geometric fidelity. This design is motivated by the observation that although point clouds are embedded in three-dimensional space, their essential geometric features often manifest distinctly along individual coordinate axes. For example, sharp edges are typically characterized by abrupt variations in two dimensions while remaining relatively stable in the third, whereas planar surfaces exhibit statistical consistency along the normal direction and greater variability within tangential directions. By capturing these dimension-specific signatures through independent FCM clustering and integrating them via a principled fusion mechanism, the proposed method attains a holistic yet nuanced representation of point cloud geometry that guides effective simplification.

Importantly, this dimensional decomposition and fusion framework extends beyond computational efficiency and reflects a conceptual shift in how point cloud structure is interpreted. Rather than treating simplification as the selection of representative points from a three-dimensional set, the problem is reframed as identifying points that simultaneously encode salient distributional characteristics across all coordinate axes. This perspective enables superior preservation of geometric features while maintaining scalability, thereby addressing a long-standing challenge in large-scale point cloud processing.

The organization of this paper is as follows: Section 2 introduces the proposed algorithm and its theoretical foundations. Section 3 details the experimental setup and a comparative analysis of results. Finally, Section 4 concludes the paper with a summary of findings and future research directions.

In this section, we develop point cloud simplification method based on Multi-dimensional FCM (MFCM) membership functions. The proposed algorithm consists of four main stages: single-dimensional FCM clustering, multi-dimensional membership fusion, adaptive sampling strategy and parameter selection.

In this study, we compare the proposed MFCM algorithm with four state-of-the-art point cloud simplification methods: Voxel Grid Down-sampling (VG), a widely adopted uniform approach that partitions space into regular voxels and retains centroid points; Random Sampling (RS), a baseline technique that randomly selects points according to a predefined ratio; Farthest Point Sampling (FPS), an iterative method that maximizes spatial coverage by sequentially selecting points farthest from existing samples; and Curvature-based Sampling (CS), a geometry-aware strategy that prioritizes high-curvature regions to preserve salient geometric features. The experiments utilized six point cloud datasets from Stanford University (Horse, Skull, Bunny, Cactus, Elephant, and Chair), evaluating performance through two widely adopted quantitative metrics:

Chamfer Distance (CD): Measures the average bidirectional distance between point sets. Lower values indicate better overall shape preservation. C D S 1 , S 2 = 1 S 1 ∑ x ∈ S 1 min y ∈ S 2 ∥ x − y ∥ 2 + 1 S 2 ∑ y ∈ S 2 min x ∈ S 1 y − x 2 (17)

Hausdorff Distance (HD): Captures the worst-case maximum deviation. Lower values ensure no significant local distortions. H D S 1 , S 2 = max sup x ∈ S 1 inf y ∈ S 2 ∥ x − y ∥ , sup y ∈ S 2 inf x ∈ S 1 y − x (18)

Both metrics were computed using efficient kd-tree implementations for nearest neighbor searches, and values were scaled by 1,000 for readability.

The comprehensive experimental evaluation reveals several important insights regarding the performance characteristics of different point cloud simplification algorithms across diverse datasets, which are plotted in Figures 1–6. The consistent superiority of both MFCM and FPS algorithms across all datasets warrants particular attention. While FPS demonstrates exceptional performance in spatial coverage maximization through its iterative selection of farthest points, MFCM achieves even better results by incorporating multi-dimensional distribution characteristics.

The performance difference can be attributed to several fundamental factors: First, FPS operates purely on spatial distances without considering the underlying data distribution patterns. While effective for ensuring even spatial coverage, this approach may oversample homogeneous regions while potentially down-sampling areas with complex geometric features. In contrast, MFCM analyzes the statistical distribution of points along each dimension independently, enabling it to identify points that are representative not just spatially but also in terms of their distributional characteristics. Second, the dimensional decomposition strategy of MFCM provides a more nuanced understanding of point importance. By examining how points contribute to the overall distribution along each coordinate axis, MFCM can better preserve structural characteristics that might be overlooked in purely distance-based approaches. This is particularly evident in datasets with anisotropic features or directional dependencies, where MFCM demonstrates significant advantages over FPS. Third, the fuzzy membership framework of MFCM introduces a degree of tolerance to noise and outliers that enhances robustness in real-world datasets. While FPS can be sensitive to outlier points that artificially extend the spatial bounds, MFCM’s membership-based approach naturally weights points according to their representativeness within clusters, providing inherent robustness.

Furthermore, the experimental results also highlight the scalability of these algorithms with respect to dataset size. For smaller datasets, both FPS and MFCM exhibit comparable performance in terms of simplification accuracy and computational efficiency. However, as the dataset size increases, MFCM demonstrates superior scalability due to its dimensional decomposition strategy, which allows for parallel processing along each dimension. This parallel processing capability significantly reduces the overall computational time, making MFCM more suitable for large-scale point cloud simplification tasks.

In addition to scalability, the experiments also investigate the impact of different parameter settings on the performance of both algorithms. For FPS, the number of iterations and the initial point selection strategy are found to have a significant impact on the final simplification result. In contrast, the performance of the developed MFCM is more stable across different parameter settings, thanks to its adaptive membership assignment mechanism. This mechanism automatically adjusts the importance of each point based on its distributional characteristics, reducing the need for extensive parameter tuning.

In summary, the experimental results not only validate the effectiveness of the proposed MFCM algorithm but also contribute to a deeper understanding of the relative strengths and limitations of different simplification paradigms. The consistent superiority of MFCM across diverse datasets and evaluation metrics suggests that its multi-dimensional, distribution-aware approach represents a significant advancement in point cloud simplification methodology.

This paper has presented a novel multi-dimensional FCM-based approach for LiDAR point cloud simplification that effectively addresses the critical trade-off between computational efficiency and geometric fidelity preservation. By decomposing the complex three-dimensional simplification problem into independent one-dimensional analyses, our method achieves significant advantages over conventional techniques.

The central innovation of the proposed approach lies in its dimensional decomposition strategy, which enables the efficient application of FCM clustering to individual coordinate axes. This design facilitates the extraction of informative membership functions that explicitly characterize the statistical distribution of point cloud data along each spatial dimension. By subsequently fusing these dimension-wise membership values using an optimized product-based strategy, a unified importance metric is constructed that effectively identifies points most representative of the underlying geometric structure.

Extensive experiments conducted on a diverse set of datasets, including benchmark three-dimensional models, real-world LiDAR scans, and architectural point clouds, demonstrate the superior performance of the proposed MFCM algorithm. In comparisons with four representative state of the art simplification methods, MFCM consistently achieves the lowest Chamfer Distance and Hausdorff Distance at identical simplification ratios. These quantitative advantages are further supported by qualitative visual evaluations, confirming that the proposed method preserves both global shape integrity and fine-scale geometric details more effectively than existing approaches.

Several promising avenues for future research naturally arise from this work. First, the computational efficiency of MFCM can be further enhanced through parallel implementations, as the dimensional independence of the framework is inherently well suited to parallel processing. Second, the proposed formulation may be extended to incorporate additional point attributes beyond spatial coordinates, such as intensity, color, or temporal information in dynamic point clouds. Third, the integration of deep learning techniques offers the potential to automate parameter selection and to learn application specific fusion strategies. Finally, a hierarchical extension of the framework would allow the algorithm to scale efficiently to ultra large point clouds containing billions of points.

In conclusion, the multi-dimensional FCM approach represents a significant advancement in point cloud simplification methodology. By fundamentally rethinking the problem through dimensional decomposition and fuzzy membership analysis, we have developed an algorithm that not only outperforms existing methods but also provides a flexible framework for future extensions and applications. As LiDAR technology continues to advance and generate increasingly large and complex point cloud datasets, methods like MFCM will play an increasingly important role in enabling efficient processing while preserving the rich geometric information essential for accurate perception and analysis.