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Geometry vs Structure: Graph-Based Diagnostics for LiDAR Point-Cloud Simulation Fidelity

Ghazal Farhani, Taufiq Rahman

arXiv:2609.16378v1cs.ROcs.CV

TL;DR

Geometric metrics for LiDAR point clouds do not explicitly capture connectivity, topology, or object-level organization, complicating fidelity assessment. This paper builds graphs from real and simulated scans, detects and matches communities, and compares bounded spectral structure with geometric similarity. Experiments report complementary information from the two metric types, with rλ robust to rigid transforms and sensor noise but sensitive to structural deformation.

  • Problem

    Point-cloud geometric metrics do not explicitly represent connectivity, topology, or object-level organization, leaving structural fidelity difficult to assess.

    Source: §I · p. 1

  • Method

    The framework builds point-cloud graphs, detects spatially coherent communities, matches them by centroid proximity, and compares matched structures using bounded spectral metrics alongside geometric measures.

    Source: §I · p. 1, 1) Community Match Rate · p. 3

  • Results

    Controlled perturbations showed rλ invariant to rigid transforms and sensor noise but sensitive to structural deformation, while CDC and rλ captured complementary aspects of fidelity.

    Source: §V · p. 8

  • Takeaways & Limitations

    Jointly reporting rλ and CDC provides a richer diagnostic for virtual-testing-environment validation than either metric alone.

    Source: §V · p. 8

Abstract

from arXiv · show

Digital twins provide a scalable and cost-effective complement to real-world testing for validating autonomous-driving and advanced driver-assistance system (ADAS) sensor pipelines. However, quantifying their fidelity remains challenging, particularly for 3D LiDAR point clouds, where conventional geometric metrics may overlook important structural discrepancies. We present a graph-based framework for evaluating the structural fidelity of simulated LiDAR point clouds against real-world scans. While scan-level metrics such as Chamfer distance capture point-wise geometric similarity, they do not explicitly represent connectivity, topology, or object-level organization. Our framework constructs graphs from real and simulated point clouds, applies Louvain community detection to identify spatially coherent subgraphs, and matches corresponding communities using centroid proximity. For each matched pair, we compute $r_\lambda$, a bounded graph-spectral metric motivated by Weyl's inequality, and compare it with density-aware Chamfer distance (CDC) as a geometric baseline. Controlled perturbation experiments demonstrate that $r_\lambda$ is invariant to rigid transformations and robust to sensor noise while remaining sensitive to structural deformation. We evaluate the framework on 50 paired real and simulated LiDAR scans acquired using a Velodyne VLP-32C sensor and CARLA, respectively. The dataset contains more than 1,000 matched communities across four representative classes: vehicles, vegetation, trees, and building walls. The results show that geometric and structural measures capture complementary aspects of simulation fidelity, supporting graph-spectral analysis as an additional diagnostic layer for validating digital twins in ADAS and autonomous-driving applications.

I. INTRODUCTION

Existing point-cloud validation metrics quantify geometric differences but often miss structural discrepancies and cannot identify which objects fail. This work adds graph-based, object-level structural analysis to geometric validation of simulated LiDAR scans.

I. INTRODUCTION

  • Global geometric metrics for point clouds cannot identify which specific objects fail fidelity requirements or diagnose structural failures.The paper treats geometric and structural fidelity as complementary properties.Source: §I · p. 1
  • Graph communities and Laplacian eigenvalue comparisons provide object-level structural diagnostics with theoretical bounds on divergence and principled threshold selection.The framework represents points as graph nodes and spatial proximity as edges.Source: §I · p. 1
  • Heat-kernel node selection reduces structurally influential points to address community-size mismatch from real and simulated sensor-density differences.The framework is evaluated through controlled perturbations and real-versus-simulated comparisons using CARLA and a Velodyne VLP-32C.Source: §I · p. 1

II. RELATED WORK

  • Chamfer Distance and EMD quantify point-wise geometric differences, but neither provides theoretical bounds on structural divergence; EMD’s O(n^3 log n) complexity is impractical for large automotive point clouds.Prior work also identifies structural properties—connectivity, topology, and relationships between object components—as outside these distance metrics’ scope.Source: §II · p. 2
  • CDC transforms point-wise Chamfer distances with an exponential kernel into [0,1], making it less sensitive to extreme outlier distances than Chamfer Distance.Source: §II · p. 2
  • The paper positions graph-based structural metrics as complementary to geometric validation, adding object-level diagnostics and theoretical guarantees on topological divergence.Source: §II · p. 2

III. GRAPHS AS METRICS

The framework represents each LiDAR point cloud as a weighted graph whose edges encode local spatial proximity. It uses true distances as edge weights and analyzes the normalized Laplacian spectrum to represent connectivity and structure.

A. From Point Clouds to Graphs

  • Each point becomes a node, and edges are the union of radius-threshold and k-nearest-neighbor connections in an undirected weighted graph.Source: A. From Point Clouds to Graphs · p. 2
  • True metric distances preserve a monotonic, geometrically interpretable penalty for increasing separation, unlike similarity kernels that compress local distance variation.Source: A. From Point Clouds to Graphs · pp. 2–3
  • The normalized graph Laplacian is symmetric positive semidefinite, and its spectrum encodes connectivity; the spectral gap is sensitive to global connectivity.Source: A. From Point Clouds to Graphs · p. 3

B. Object-Level Decomposition via Communities

The framework uses Louvain modularity maximization to partition point-cloud graphs into communities, then compares real and simulated scenes through community matching and within-object metrics. Centroid proximity supports matching even when the detected community counts differ.

B. Object-Level Decomposition via Communities

  • Louvain modularity maximization partitions each graph into densely connected communities, yielding community sets for real and simulated point clouds.The modularity objective uses edge count, weighted degree, and whether node pairs share a community.Source: B. Object-Level Decomposition via… · p. 3

C. Graph-Based Validation Metrics

  • The framework reports scene/object presence through community matching and within-object structural similarity through bounded spectral and diffusion indicators.Source: C. Graph-Based Validation Metrics · p. 3
  • Centroid proximity matches communities even when real and simulated scenes contain different numbers of objects; unmatched communities indicate missing or hallucinated structures.Source: 1) Community Match Rate · p. 3

2) Density-Aware Chamfer Distance (CDC):

The section pairs CDC as a geometric baseline with graph-spectral indicators of structural change and heat-kernel influence.

2) Density-Aware Chamfer Distance (CDC):

3) Heat Kernel and Influence:

  • Heat-kernel row sums define node influence, emphasizing nodes that are structurally central under diffusion.The heat kernel models diffusion on the graph with diffusion time t > 0.Source: 3) Heat Kernel and Influence · p. 3
  • rgap uses a Weyl-based bound on spectral-gap deviation and is sensitive to connectivity-regime changes and near-disconnectivity.The gap deviation is bounded by 2∥∆L∥2.Source: 4) Bounded Gap Change (rgap) · p. 3
  • rλ measures the largest normalized eigenvalue shift across the full Laplacian spectrum as a bounded indicator of within-object structural change.It compares reduced normalized Laplacians after heat-kernel node selection and spatial alignment.Source: 5) Bounded Spectral Change (rλ) · p. 3

D. Practical Pipeline

The practical pipeline matches real and simulated communities, reduces density-mismatched pairs to equal size, and computes per-community metrics.

D. Practical Pipeline

1) Community Matching:

2) Heat-Kernel Node Selection for Density Mismatch:

E. Data Collection and Graph Build-up

The study builds a CARLA virtual testing environment from registered LiDAR data, then constructs object-focused graphs from real and simulated scans.

E. Data Collection and Graph Build-up

  • Synchronized LiDAR, IMU, and camera measurements support odometry estimation and registered point clouds used to build a 3D mesh imported into CARLA.The resulting environment is one-to-one with the real scene geometry for direct scan comparison.Source: E. Data Collection and Graph Build-up · p. 4
  • Each scan graph connects points to their k nearest neighbors subject to a radius threshold, and community detection is applied to real and simulated graphs.A Z-value cutoff excludes the road surface so graph comparisons focus on objects above ground.Source: E. Data Collection and Graph Build-up · p. 4
  • PCA iteratively adjusts the Z-threshold so variance loss from the cutoff remains within 1% of total variance.Ground removal is widely used [18], with similar PCA-based road-surface approaches introduced in.Source: E. Data Collection and Graph Build-up · p. 4

A. Controlled Perturbation Analysis

Controlled perturbations test metric responses across four object classes, with results distinguishing rigid pose changes from noise-driven structural changes.

A. Controlled Perturbation Analysis

1) Rigid Transforms:

  • Translations of 0.05 m and rotations of 2° yield rλ < 0.01 and rgap < 0.0001 across all four classes.The authors attribute this invariance to rigid transforms preserving point-proximity relationships and leaving the graph Laplacian unchanged.Source: 1) Rigid Transforms · p. 4

2) Gaussian Noise:

  • For ε ≤10−2, metrics remain small across classes, while car rλ jumps between ε = 10−2 and 10−1 as noise rewires its curved surface.The tree response is near-flat, while vegetation grows slowly and the wall shows a moderate steady rλ increase.Source: 2) Gaussian Noise · p. 4, Figure 2 · p. 5

A. Controlled Perturbation Analysis

  • Under point dropout, car and vegetation are most sensitive, while wall is most robust because of connectivity redundancy in planar structures.Fig. 2(d,e) presents the dropout effects.Source: Figure 2 · p. 5

3) Point Dropout:

Point dropout disrupts community matching and structural metrics more than Gaussian noise or small rigid transforms. In real-versus-simulated scans, geometric and graph-spectral measures often diverge, while unmatched communities reveal missing or hallucinated scene structures.

3) Point Dropout:

  • Dropout affected all object classes more strongly than noise: cars and vegetation reached rλ ≈ 0.40 and 0.35 at 40% dropout, while wall rgap remained below 0.004.The authors attribute wall robustness to redundant paths in planar structures; rλ captures distributed internal deformation, whereas rgap is more sensitive to bottlenecks or near-disconnections.Source: 3) Point Dropout · p. 5
  • Because simulated-to-real community density ratios ranged from below one quarter to above four, heat-kernel selection reduces each matched pair to nkeep = min(nreal, nsim) before spectral comparison.The reduction is intended to prevent density mismatch from confounding structural comparison.Source: 3) Point Dropout · p. 5
  • For Gaussian noise, matched-community rates stayed above 95% even at the largest tested level, although centroid distance and eigenspectrum distance rose gradually.The results indicate that community structure was largely preserved under Gaussian noise.Source: 3) Point Dropout · p. 5
  • Gaussian noise retained a community match rate above 95%, small rigid transforms retained 100%, and dropout reduced the rate from approximately 91.5% at p = 0.05 to 85.4% at p = 0.40.For rigid transforms, dλ stayed essentially negligible despite increased centroid distance; dropout increased both centroid distance and dλ.Source: 4) Global Point Cloud Perturbation · p. 5, 3) Point Dropout · p. 5

B. Real experiments

  • Across 50 scan pairs, the method found 1,067 matched communities with a one-to-one match rate of 59.4%±0.066.Unmatched real communities were often distant from the sensor, while unmatched simulated communities included hallucinated structures, especially dense foliage and distant objects.Source: B. Real experiments · p. 6
  • Across 50 scans, CDC averaged 0.75±0.02, rλ averaged 0.37±0.03, and rgap averaged 0.028±0.014; rλ varied more across matched objects than CDC.The small rgap indicates that matched communities generally retained their coarse connectivity regime, while rλ revealed internal structural changes.Source: B. Real experiments · p. 6
  • Across four scans, CDC stayed high and weakly dispersed while rλ varied more; for a vehicle, CDC = 0.56 and rλ = 0.43 indicated moderate differences.The vehicle values show an instance where the geometric and structural measures agreed reasonably well.Source: B. Real experiments · pp. 6–7

1) Pipeline Timing:

The pipeline evaluates scan pairs offline, with an end-to-end runtime of approximately 7,120 ms per pair on a high-RAM CPU runtime.

1) Pipeline Timing:

  • Approximately 7,120 ms was required for the full end-to-end pipeline per scan pair.The authors consider the overhead acceptable because fidelity evaluation is performed offline.Source: 1) Pipeline Timing · p. 7
  • Community detection took 544 ms, Laplacian computation 596 ms, heat-kernel computation 1,505 ms, and Hungarian spatial alignment 1,532 ms per scan.These are the four primary pipeline-stage runtimes reported.Source: 1) Pipeline Timing · p. 7
  • Experiments ran on Google Colab's high-RAM CPU runtime and required no specialized hardware.The reported runtime includes intermediate steps beyond the four primary stages.Source: 1) Pipeline Timing · p. 7

C. Ablation Study

The ablation selected graph parameters and found diffusion time had little effect. The conclusion presents CDC and rλ as complementary diagnostics, with rλ responding to structural artifacts and the metrics often diverging for trees and vegetation.

C. Ablation Study

V. CONCLUSION

  • CDC measures geometric similarity whereas rλ measures structural similarity, reflecting distinct fidelity dimensions that can diverge in connectivity and geometric placement.Source: §V · pp. 7–8
  • rλ resisted rigid transforms and sensor noise but responded to structural deformation from point dropout and hallucination; low real-data match rates largely reflected simulator deficiencies.Source: §V · p. 8
  • CDC stayed consistently high with limited discrimination, while rλ had a larger dynamic range; the metrics often decoupled for trees and vegetation but were generally consistent for vehicles.Source: §V · p. 8
  • The framework is model-free and requires no labeled data; the authors propose reporting rλ with CDC to reveal failure modes neither captures alone.Source: §V · p. 8

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