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Superquadric Primitive Decomposition of 3D point clouds via Geometric-Aware Inlier Refinement

Alessandro Rinaldi, Edoardo Tedesco, Andrea Ferraris, Filippo Leveni, Daniele Baieri, Filippo Maggioli, Simone Melzi, Luca Magri

arXiv:2609.35725v1cs.CV

TL;DR

3D primitive fitting is challenging when residual and proximity cues misassign points across adjacent surfaces. The paper adds normal-aware inlier refinement through graph-cut energy minimization in single- and multi-model pipelines. Experiments report faster execution than GC-RANSAC alongside consistently more accurate decompositions.

  • Problem

    Residual- and proximity-based methods can assign points incorrectly across adjacent or overlapping primitives, complicating robust primitive decomposition.

    Source: §1 · p. 1, §3 · p. 4

  • Method

    GAIR formulates normal-aware inlier refinement as graph-cut energy minimization and integrates it into single- and multi-model estimation.

    Source: §1 · p. 2, §8 · p. 18

  • Results

    GAIR remains faster than GC-RANSAC while consistently providing more accurate decompositions.

    Source: §8 · p. 19

  • Takeaways & Limitations

    The formulation provides a foundation for incorporating additional geometric constraints beyond the normal consistency emphasized here.

    Source: §8 · p. 18

    Compact superquadrics smooth fine-scale shape details, with the largest errors at high-curvature regions such as the beak; more primitives could recover such detail.

    Source: §5.6 · p. 15

Abstract

from arXiv · show

The decomposition of 3D point clouds into interpretable geometric primitives remains a longstanding challenge in Computer Vision and Computer Graphics. Among the available representations, superquadrics offer a compact and expressive model capable of capturing a wide range of shapes. However, their estimation is inherently challenging, as it requires solving a non-linear optimization problem and is particularly sensitive to noise, outliers, and overlapping structures. While robust estimation methods such as RANSAC and its variants achieve strong performance, they rely primarily on spatial proximity and residual-based criteria, often leading to incorrect inlier assignments across adjacent or complex arrangements of primitives. In this work, we introduce a geometric-aware framework for primitive decomposition that explicitly incorporates local surface properties into the fitting process. Specifically, we propose an inlier refinement step formulated as an energy minimization problem and solved via graph-cut optimization. Our formulation integrates geometric priors, such as normal consistency, enabling more reliable inlier selection beyond purely residual-based criteria. The approach naturally applies to both single-model estimation and multi-model decomposition. By leveraging geometric information beyond point-wise residuals, our method reduces erroneous inlier propagation and stabilizes parameter estimation. Experiments on synthetic and real datasets show consistent improvements in geometric accuracy, robustness to noise and outliers, and convergence efficiency compared to state-of-the-art RANSAC-based methods.

1. Introduction

Primitive decomposition uses compact, interpretable models, but nearby or overlapping surfaces make residual- and proximity-based inlier assignment unreliable. GAIR adds local geometric cues to inlier refinement and applies the approach to single-model fitting and multi-model decomposition.

1. Introduction

  • Nearby points can belong to different surfaces, so residual- and proximity-based refinement may propagate inliers across adjacent primitives.The challenge is intensified by noise, outliers, and the joint estimation of segmentation and model parameters in scenes with partially overlapping primitives.Source: §1 · pp. 1–2
  • GAIR incorporates local surface structure, including normal consistency, into inlier refinement rather than relying only on residuals and proximity.Nearby points with inconsistent normals should not be assigned to the same structure, although similar normals alone do not establish shared primitive membership.Source: §1 · p. 2
  • The refinement is formulated as energy minimization over point labels and solved by graph cuts, adding geometric priors to consensus-maximization pipelines.The framework is intended to reduce erroneous propagation across adjacent primitives and stabilize estimated models.Source: §1 · p. 2
  • Superquadrics compactly represent varied shapes, but their nonlinear parameter estimation is sensitive to noise, outliers, and initialization.Their fitting difficulty makes them a testbed for evaluating 3D inlier-refinement strategies.Source: §1 · p. 2

1.1. Contributions.

  • GAIR-RANSAC integrates geometric-aware refinement into consensus maximization for single-model superquadric fitting, with reported gains in stability, parameter accuracy, and accuracy–runtime trade-off.The paper evaluates the approach on synthetic and real data and reports improved robustness and stability over existing methods.Source: §1.1 · p. 3
  • GAIR-RANSACOV combines GAIR-based hypothesis generation with Maximum Coverage model selection for multi-superquadric decomposition, particularly with adjacent or overlapping primitives.This extends the paper's earlier conference version to multi-model decomposition and a broader evaluation.Source: §1.1 · p. 3

2. Problem formulation

The problem is to explain an unordered, noisy 3D point set with superquadric primitives using point normals and a residual threshold. Each primitive is parameterized by its shape and pose, and the radial residual uses a projection along the ray from the primitive center.

2. Problem formulation

  • The input is an unordered set of noisy 3D points with associated normals that are provided or estimated and consistently oriented.The goal is to recover superquadric parameters for a set of primitives; their number may be specified or estimated.Source: §2 · p. 3
  • The inlier threshold ε marks points whose primitive residuals fall below it and is supplied as an estimate of expected measurement uncertainty.The formulation focuses on superquadrics but can extend to planes, spheres, or surface patches.Source: §2 · p. 3
  • A superquadric uses three axis-scale parameters and two shape exponents, with a rigid rotation and translation giving 11 shape-and-pose parameters.The exponents control roundness or sharpness; ε1 governs z-axis geometry and ε2 the cross-section in the orthogonal plane.Source: §2 · pp. 3–4
  • For residual evaluation, scene points are mapped into the superquadric's canonical coordinates, and the radial distance is measured along the ray from the primitive center.The surface point on that ray is obtained by scaling the mapped point using the implicit function.Source: §2 · p. 4
  • The formulation is geometrically intuitive and accurate for convex shapes, but less reliable for highly non-convex configurations (ε1 or ε2 much greater than 1), where projection may deviate from the true closest point.Source: §2 · p. 4

3. Related Work

Related work covers robust single-model estimation, multi-model fitting, and parametric primitive recovery. Existing approaches use residuals and spatial coherence, but proximity can mislead in adjacent or intersecting primitive scenes; this work incorporates normal information directly into inlier optimization.

3. Related Work

  • RANSAC iteratively samples minimal data subsets and selects the hypothesis with the largest consensus; LO-RANSAC refines models using current inliers, while GC-RANSAC adds graph-cut spatial regularization.Its smoothness term encourages neighboring low-residual points to be jointly selected as inliers.Source: §3 · p. 4
  • Spatially coherent refinement works when proximity reflects model membership, but can misassign points where 3D primitives are adjacent or intersecting.In those settings, spatial proximity does not necessarily reflect the underlying surface structure.Source: §3 · pp. 4–5
  • Multi-model fitting jointly estimates structures and point assignments; optimization methods balance residual fidelity with spatial regularization, while RANSACOV selects models by maximum coverage.RANSACOV can better handle overlapping structures but retains residual-based inlier definitions.Source: §3 · pp. 4–5
  • Primitive fitting approximates complex shapes with parametric forms, and superquadrics represent varied shapes with relatively few parameters.Prior work used distance and angular thresholds to validate primitive inliers, whereas this formulation places normal information in unary and pairwise optimization terms.Source: §3 · p. 5
  • Other primitive-fitting approaches include iterative recover-and-select and probabilistic EM methods for noise and outliers.Differentiable Blocks World uses multi-view images and substantial computation, while stochastic search handles noisy point clouds but relies on predefined primitive types.Source: §3 · p. 5
  • Learning-based methods infer primitive collections from clean training data but typically require large annotated datasets and may generalize poorly across domains or primitive families.This work follows an algorithmic direction and extends its earlier formulation to multi-model fitting.Source: §3 · p. 5

4. Method

GAIR refines point-cloud inliers using local surface geometry, and the paper builds on it for single-superquadric fitting and multi-primitive decomposition. The method combines energy-based labeling with graph cuts and a structured consensus pipeline.

4.3. GAIR-RANSACOV: Primitive Decomposition

  • GAIR-RANSACOV extends the framework to multiple superquadrics by generating candidate hypotheses and selecting a consistent decomposition through global Maximum Coverage.It decouples hypothesis generation from model selection; the selection objective chooses at most κ models whose union of inlier sets explains the most points.Source: §4 · p. 5, §4.3 · p. 9

4.1. Geometric-Aware Inlier Refinement

  • GAIR refines residual-thresholded inliers by removing geometrically inconsistent points and enforcing local coherence, improving model adherence and the fitting pipeline’s accuracy–runtime tradeoff.The method treats inlier membership as a binary labeling informed by surface-level properties such as aligned normals.Source: §4.1 · p. 6

4.1.1. Energy Formulation

  • The energy combines unary data fidelity based on residual and normal orientation with pairwise coherence that encourages neighboring points to share labels when their geometry aligns.The pairwise term is residual-aware, reducing the influence of points far from the model and extending proximity-based refinement with normal consistency.Source: §4.1.1 · pp. 6–7

4.1.2. Graph Construction and Optimization

  • GAIR minimizes the labeling energy using graph cuts on a sparse neighborhood graph, with nodes representing points and edges connecting local neighbors.The implementation uses k-nearest neighbors, with k=6 in controlled experiments and k=10 for denser real scans.Source: §4.1.2 · pp. 7–8

4.2. GAIR-RANSAC: Single-Model Superquadric Fitting

  • GAIR-RANSAC adapts hypothesize-and-verify fitting with locally coherent sampling, PCA-initialized nonlinear estimation, and iterative GAIR refinement followed by inner RANSAC updates.The loop ends when consensus no longer improves, returning the model with the highest refined support.Source: §4.2 · p. 8

4.3. GAIR-RANSACOV: Primitive Decomposition

  • For decomposition, sequential GAIR-RANSAC extracts hypotheses from residual data, while dilated inlier removal discourages duplicate local support and coverage validation filters weak models.Runs use random subsamples; the remaining pool can contain redundant models, so selection uses at most κ hypotheses to maximize explained points.Source: §4.3 · p. 9

5. Experiments

The experiments assess GAIR for single-model fitting and multi-model decomposition on synthetic and real point clouds, examining inlier classification, geometric fidelity, and computational behavior. They report improved robustness and reconstruction with geometric-aware refinement, including lower Chamfer Distance and Inlier Assignment Error across tested budgets for GAIR-RANSACOV.

5.1. Datasets

  • The evaluation spans single- and multi-model SqSoup, closely placed TangentSuperquadrics, irregular Thingi10K shapes, Sketchfab scans, and three LiDAR scenes.Sketchfab results are qualitative because ground-truth decompositions are unavailable; the LiDAR scenes range from separated primitives to partially overlapping or noisy, weakly separated structures.Source: §5.1 · pp. 10–11

5.2. Metrics

  • The evaluation measures inlier-outlier classification with IAE, surface fidelity with CD and HD, and computational behavior with execution time and local-optimization steps.IAE counts incorrectly classified inliers or outliers; normalized CD and HD are reported for Thingi10K and SqSoup after bounding-box normalization.Source: §5.2 · p. 11

5.3. Baselines and evaluation protocols

  • The baselines progress from residual-threshold Vanilla RANSAC to residual-based LO-RANSAC and spatial-coherence GC-RANSAC; GAIR adds normal consistency to residual and spatial information.Multi-model comparisons use sequential extraction, with a RANSACOV post-processing step selecting up to κ models; methods share a 20-iteration maximum and other controlled fitting settings.Source: §5.3 · p. 11

5.4. Sensitivity to the inlier threshold

  • GAIR-RANSAC is consistently more accurate and less sensitive to the inlier threshold than Vanilla RANSAC on TangentSuperquadrics.With Gaussian noise σ=0.4 and 10% outliers, the comparison varies ε=s·σ for s from 1 to 4; GAIR remains stable across the tested range, unlike Vanilla RANSAC.Source: §5.4 · p. 12, Table 2 · p. 12

5.5. Single-model fitting

  • In single-model SqSoup fitting, GAIR-RANSAC performs best across outlier levels, reducing IAE by approximately a factor of two versus GC-RANSAC.GAIR-RANSAC and GC-RANSAC have comparable execution times, while GAIR-RANSAC achieves lower error; LO-RANSAC is cheaper but less accurate. The tests increase uniformly distributed outliers on single-superquadric clouds.Source: §5.5 · pp. 12–13, Figure 5 · p. 13

5.6. Primitive decomposition

  • On Tangent Superquadrics, GAIR-RANSAC consistently outperforms GC-RANSAC across noise and outlier conditions in CD, HD, and IAE, with lower result variance.The benchmark stresses adjacent and touching primitives, where Euclidean proximity is unreliable for assigning points to models.Source: §5.6 · pp. 13–14, Table 4 · p. 13
  • Across ten SqSoup and 3D Shapes point clouds, GAIR-RANSACOV achieves the lowest Chamfer Distance at every tested outlier level; its IAE is also lowest at every nonzero level except clean data.At low outlier levels, methods have similar IAE despite differing Chamfer distances; GAIR’s IAE advantage grows with contamination.Source: §5.6 · pp. 14–15, Table 5 · p. 14
  • Qualitative examples show coherent decomposition under severe outlier contamination, while the Bird example smooths fine details and has its largest errors around distinctive high-curvature regions such as the beak.The section also describes a fixed-run-time comparison with 60-, 90-, or 120-second candidate-generation budgets and RANSACOV selection; no outcome for that comparison appears in the supplied passages.Source: Figure 7 · p. 14, §5.6 · p. 15
  • GAIR-RANSACOV achieves the lowest Chamfer Distance and Inlier Assignment Error at every tested budget, improving with more computation while competing methods tend to plateau.Results average all ten point clouds from SqSoup and 3D Shapes over 16 independent trials.Source: §5.6 · p. 15
  • On Sketchfab scans, GC-RANSAC and GAIR-RANSACOV share the same set-cover stage, isolating inlier refinement as the source of final decomposition differences.Figure 9 compares the input point clouds with decompositions from both methods.Source: §5.6 · p. 15, Figure 9 · p. 16
  • By limiting propagation across surface discontinuities, GAIR-RANSACOV generates part-aligned candidates, while GC-RANSACOV's merged hypotheses leave components unrecoverable by set-cover and cause under-segmentation.GAIR-RANSACOV produces decompositions more consistent with geometric parts, whereas GC-RANSACOV systematically merges distinct components.Source: §5.6 · pp. 15–16

6. Ablation studies on the GAIR energy

The Hammer ablation compares energy-term variants and finds that the complete GAIR formulation combines the lowest average error with the most stable results. The tests also show that normal-aware pairwise refinements are sensitive to normal-estimation errors.

6. Ablation studies on the GAIR energy

  • GAIR-U performs comparably to GC-RANSAC, so replacing only the unary term with its normal-aware counterpart does not substantially improve reconstruction.Source: §6 · p. 17
  • GAIR-Uniform achieves similar average error while reducing run-to-run variability, consistent with its residual-aware pairwise term providing more stable regularization.Source: §6 · p. 17
  • GAIR-Hard lowers average error but has substantially higher variability because binary thresholding gives equal weight to retained edges, including marginal ones.Small changes in estimated normals or sampled hypotheses can therefore markedly change label propagation.Source: §6 · p. 17
  • The complete GAIR formulation achieves the lowest average error and most stable results by removing incompatible edges and weighting retained connections by normal coherence.The study evaluates energy-term variants on the Hammer point cloud using eight independent runs per configuration.Source: §6 · pp. 16–17, Figure 10 · p. 17
  • As normal angular error increases, the advantage of normal-aware variants decreases, and pairwise variants are most affected because corrupted normals alter graph connectivity and edge weights.When normals are severely inaccurate, formulations that do not rely on them are preferable.Source: §6 · p. 17

7. Primitive decomposition of real LiDAR scans

On real scans with acquisition artifacts, GAIR-RANSAC better preserves boundaries and decomposes adjacent parts more completely in challenging scenes. Fine part-level decomposition remains difficult when scans lack surface coverage or clear geometric separation.

7. Primitive decomposition of real LiDAR scans

  • The real-scene tests include three scans with non-uniform sampling, missing regions, and imperfect surface coverage; normals are estimated from the noisy acquired geometry.Dense scans are uniformly subsampled before decomposition.Source: §7 · p. 17
  • For the simple ball-and-box scene, both methods recover the expected two-primitive decomposition despite occlusion of the box's lower face.Source: §7 · p. 17, Figure 11 · p. 18
  • On the sofa, GAIR-RANSAC better preserves cushion boundaries and produces a more complete decomposition than GC-RANSAC, which propagates inliers across contacts and misses sparse lower components.Source: §7 · pp. 17–18, Figure 11 · p. 18
  • On the plush toy, GAIR-RANSAC separates the head, torso, limbs, and muzzle more consistently with visible structure, though it misses some fine details.GC-RANSAC instead merges the head and torso into one large superquadric; the toy has fuzzy surfaces, weakly separated parts, and noisy normals.Source: §7 · p. 18, Figure 11 · p. 18
  • Across noisy, incomplete LiDAR scans, GAIR improves decomposition for objects with adjacent parts, but fine part-level decomposition remains difficult with insufficient coverage or unclear boundaries.Source: §7 · p. 18

8. Future Work and Conclusions

The conclusion presents GAIR as a geometric-prior extension to GC-RANSAC that works in single- and multi-model pipelines, with more accurate decompositions and faster execution than GC-RANSAC in the experiments. Future work proposes additional geometric constraints, global multi-model fitting, and automatic selection of primitive count.

8. Future Work and Conclusions

  • Future work proposes adding a Chamfer-distance term to favor compact superquadrics that cover point data without extending excessively into empty regions.Source: §8 · p. 18
  • A global multi-label optimization, with labels for models and outliers, is proposed to reduce dependence on model-extraction order and potentially improve robustness in complex scenes.Source: §8 · p. 18
  • The current pipeline requires the number of primitives to be specified in advance; future work proposes incorporating model-complexity selection into RANSAC.Source: §8 · p. 18
  • GAIR-RANSAC and GAIR-RANSACOV incorporate geometric priors into single- and multi-model estimation, and GAIR is faster than GC-RANSAC while providing more accurate decompositions in the experiments.Source: §8 · pp. 18–19
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