Source-linked AI summary
Bayesian methods and Markov chain Monte Carlo algorithms for curve reconstruction and point cloud data analysis
Asir Intesar Tushar, Ioannis Sgouralis
TL;DR
Point-cloud reconstruction must handle large datasets, noise, missing information, and absent uncertainty quantification. The paper introduces a fully Bayesian model for noisy observations constrained to closed curves, with non-parametric regularization and specialized MCMC inference. Experiments report accurate reconstructions and uncertainty quantification, while the method remains computationally costly and is tailored to one-dimensional geometries.
Problem
Point-cloud analysis is hindered by large data volumes, localization noise, missing points, and reconstruction pipelines that lack uncertainty quantification.
Method
The framework models noisy observations around latent positions on a closed curve, regularizes geometry with a non-parametric prior, and uses specialized MCMC samplers for posterior inference.
Results
The framework accurately reconstructs complex and polygonal curves, with posterior samples concentrated around ground truth and Sampler 3 achieving Hausdorff distance 0.075.
Takeaways & Limitations
The approach supports curve reconstruction and uncertainty quantification from incomplete, sparse, or degraded point-cloud observations.
Takeaways & Limitations
MCMC computation can remain substantial, and the current design is tailored to curves rather than surfaces or volumetric structures.
Abstract
from arXiv · showhide
Point-cloud data routinely captured by modern imaging and sensor technologies provide detailed geometric descriptions of objects and environments, but their analysis is hindered by large data volumes, localization noise, and missing information. In addition, existing point-cloud reconstruction pipelines typically return a single best-fit structure without uncertainty quantification. We introduce a fully Bayesian framework for representing point-cloud data and reconstructing closed curves, in which observed points are modeled as noisy perturbations of latent locations constrained to lie on the underlying curve that is regularized by a non-parametric prior. Posterior inference in our framework is carried out using a series of Markov chain Monte Carlo samplers tailored to point-cloud characteristics. Numerical experiments, including synthetic examples and real-world LiDAR datasets, show accurate reconstructions and quantified uncertainty over the recovered curves.
1 Introduction
Point-cloud analysis is challenged by scale, localization noise, missing observations, and limited uncertainty quantification. The study addresses these issues with a fully Bayesian framework for curve reconstruction and efficient evaluation methods.
- Large point-clouds impose high computational costs, while sensor constraints introduce localization noise and missing points.
- Existing reconstruction pipelines may omit uncertainty quantification, limiting accurate estimation of recovered structures.
- Bayesian methods incorporate uncertainty and prior knowledge to support probabilistic inference for corrupted or incomplete point-clouds.
- The proposed fully Bayesian method combines statistical curve reconstruction, noisy-data representation, and efficient computational procedures.
- The framework is evaluated on simulated and real-world data, including applications to curve reconstruction in multidimensional point-clouds.
2 Methods
The methods represent noisy point-cloud localizations as latent positions on a closed polyline, regularize its geometry with a non-parametric prior, and approximate inference using specialized MCMC sampling. The formulation supports curve reconstruction and uncertainty quantification while using a finite approximation for computational tractability.
- 2.1 Point-cloud representation: The model treats each observed localization as a noisy perturbation of a latent position constrained to an underlying closed curve.Noise is modeled with an isotropic Gaussian distribution.
- 2.2 Geometry representation: The curve is represented as a closed polyline whose active vertices are selected from inducing points, with the final segment connecting the last active vertex back to the first.The first B inducing points determine the geometry.
- 2.1 Point-cloud representation: Each latent position is assigned to a curve segment using a discrete segment indicator and a continuous local coordinate.The model estimates B, inducing-point geometry, segment indicators, local coordinates, and noise precision.
- 2.2 Geometry representation: A hierarchical construction makes latent positions uniformly distributed along the curve while accounting for segment lengths and cumulative arc-length.The resulting global coordinate is uniform over the curve’s total length.
- 2.3 Bayesian considerations and non-parametric prior: The non-parametric prior uses infinitely many candidate inducing points but keeps the number of active vertices finite to avoid overfitting.The active vertices are the first B inducing points used in the geometry.
- 2.4 Parametric model approximation: Because the infinite formulation is computationally intractable, inference uses a finite large-K approximation whose distribution converges to the non-parametric model as K →∞.The finite approximation preserves the expected number of active inducing points, E[B] = γ.
- 2.5 Markov chain Monte Carlo: The posterior is characterized by specialized MCMC samplers because it lacks a convenient analytical form and naive local updates mix slowly.The sampler sequence introduces increasingly structured updates that marginalize progressively larger variable blocks.
3 Results
Synthetic and real-world analyses show that the Bayesian framework reconstructs complex and polygonal curves while quantifying how uncertainty changes with data quality and sampling strategy.
- Synthetic validation: The framework was validated on synthetic curves of varying complexity using posterior samples and MAP reconstructions compared with ground truth.Accuracy was quantified with the Hausdorff distance between each posterior sample and the ground-truth curve.
- Synthetic validation: For a self-intersecting Lissajous curve, MAP and posterior reconstructions closely agreed with the ground truth, with low global Hausdorff discrepancy.The posterior over active vertices indicated that the model adjusted the geometry to match the intricate curve.
- Synthetic validation: For a regular octagon, MAP and posterior reconstructions closely matched the ground truth despite discontinuities at polygonal corners.This provided a simpler benchmark for recovering edges and vertices.
- Uncertainty propagation: With fixed noise τ = 30, reducing point-cloud size from N = 150 to 100 and 50 widened the posterior and increased reconstruction uncertainty.The sensitivity analysis used a regular pentagon dataset with σ = .18.
- Uncertainty propagation: With N = 1000, increasing noise from τ = 3000 to 30 and 0.3 widened the posterior and increased uncertainty over the reconstructed curves.The corresponding standard deviations were σ ≈0.018, 0.18, and 1.8.
- Comparison of sampling schemes: Sampler 3 recovered the ground truth more accurately than Samplers 1 and 2 on the same synthetic point-cloud.The MAP Hausdorff distances were 0.312, 0.318, and 0.075 for Samplers 1, 2, and 3, respectively; specialized updates helped Sampler 3 avoid partially matching configurations.
- Real-world and benchmark data: On standardized point-clouds, the framework produced posterior ensembles rather than only deterministic point estimates and reconstructed sparse, non-uniform LiDAR structures coherently.The LiDAR case studies showed uncertainty increasing in undersampled and high-curvature regions.
4 Discussion
The framework combines probabilistic modeling, efficient computation, and posterior uncertainty quantification for curve reconstruction from incomplete or degraded point-clouds. Its main limitations are substantial MCMC cost and a design currently tailored to one-dimensional geometries.
- The framework integrates latent-geometry modeling, localization-noise representation, and computational strategies for reconstructing curves with uncertainty quantification.
- Unlike deterministic pipelines, it produces posterior distributions over admissible reconstructions, enabling credible intervals, probabilistic error bars, and uncertainty-aware decisions.
- The computational cost of repetitive MCMC sampling may bottleneck large point-clouds or time-critical applications.
- The current computational design targets curves embedded in 2D, 3D, or higher-dimensional Euclidean scenes and does not directly extend to surfaces or volumes.Higher-dimensional latent objects and likelihood integrals become markedly more complex.
- 3,000 MCMC iterations with 50% burn-in and 1:8 thinning were used for each field LiDAR reconstruction.
- Parallel and distributed computing could reduce MCMC burden by exploiting likelihood independence over data points and segments.
A Main posterior factorization and definitions
The posterior target is represented through factorized conditional components, with constants depending on the curve geometry. Because one direct computation is numerically unstable, the implementation uses a robust right-hand-side expression.
- The model’s conditional posterior is sampled using a factorization for most targets in section B.
- A Binomial distribution is used for B with parameters K and γ/K.
- The joint term n(un, sn) is factorized into conditional components for wn, un, and sn given geometry G and noise τ.
- The factorization’s constants depend implicitly on the geometry G = {B, h1:K}.
- One target requires computing a sum that is prone to catastrophic cancellation and numerical underflow.
B.1 Target in Updates A and A*
The target in Updates A and A* is sampled through a defined computational procedure. The supplied passages identify the derivation and sampling step but do not provide further mathematical detail.
- The target for Updates A and A* is derived according to the procedure introduced in this section.
- This target is sampled via DS, with its constants defined afterward.
B.2 Target in Updates B, C, D, and B*
The target is derived using the factorization in section A and sampled via DS, with Normal[0,1] denoting a truncated normal distribution.
- The target is derived from the factorization in section A.
- The resulting target is sampled via DS.
- Normal[0,1] denotes the truncated normal distribution.
B.3 Target in Updates B, C, D, and B*
The target is again derived using the factorization in section A and sampled via DS.
- The target is derived from the factorization in section A.
- The resulting target is sampled via DS.
B.4 Target in Update B
The completed target is sampled through two Gibbs stages, updating b separately and then jointly updating B and h1:K using specialized samplers and proposal mechanisms.
- The target is completed with a discrete auxiliary variable b.
- Two Gibbs stages update b, followed by a joint update of B and h1:K.
- The b update targets p(b|B, h1:K, τ, w1:N) and is sampled with DS.
- The joint B and h1:K update is factorized and sampled with AS, while its last factor is sampled with ESS.
- A custom MH sampler mixes two proposal distributions, Q1 and Q2, with equal probability.
- Q1 applies a circular shift and reconnection, whereas Q2 uses reversal, reflection, and the same reconnection operation.
- Both proposals are symmetric MCMC proposals, and the acceptance reduces to that of the Metropolis sampler.
- The target includes an integral and constants ϕ′ and β′ specified as in section B.1.
B.10 Target in Update B*
Update B* uses auxiliary-variable completion and a custom MH step to modify geometry by adding or removing vertices. The proposal structure simplifies the acceptance calculation to a Metropolis form.
- The target is completed with auxiliary variables g1, g2, σ, and γ.
- The completed target is sampled with two Gibbs stages, including a joint update of B and h1:K.
- The MH proposals use a distributional family that allows factorization of the proposal.
- Further assumptions are imposed on proposal factors to achieve add/remove proposals with high acceptance rate.
- Add proposals in G correspond to remove proposals in F, while dives-in in G correspond to dives-out in F.
- The function R attracts or expels h′′′ according to γ and σ around an anchor determined by h′, h′′, g1, and g2.
- The add/remove mechanism changes B and vertex positions through proposals whose acceptance ratio is computed from target and proposal ratios.
- Because atoms other than Bold and Bprop do not move, a factor drops out and MH acceptance reduces to Metropolis acceptance.
C Additional results
Additional experiments evaluate the reconstruction framework on synthetic Lissajous and Sawtooth curves and the standardized Bottle example. The Sawtooth ground truth combines smooth and non-smooth segments in a closed counterclockwise curve.
- Synthetic examples: The Lissajous experiment uses a noisy point cloud with N = 1000 and noise τ = 500 (σ ≈0.045).The reconstruction runs for 3,000 MCMC iterations with 30% burn-in and 1:3 thinning for visualization.
- Synthetic examples: The Sawtooth experiment evaluates reconstruction on a semicircle with a zigzag lower side.Its inducing points are ordered to produce a closed counterclockwise curve.
- Synthetic examples: The Sawtooth ground truth contains both smooth and non-smooth segments and resembles the standardized Sawtooth example from Ref. [31].
- Standardized example: The Bottle experiment evaluates reconstruction on the standardized Bottle example from Ref..It uses 3,000 MCMC iterations, 50% burn-in, and 1:5 thinning for visual purposes.
- Synthetic examples: The Sawtooth reconstruction uses 3,000 MCMC iterations, discards 20% as burn-in, and applies 1:3 thinning for visualization.